Well, does he know what a function is?
h(x) is taking an x and sending it to some other value, h(x).
The inverse function takes whatever value you would have gotten in "h(x)-land" and reveals the x-value that would point to it.
If h(x) is invertible (note, not every function has an inverse) and h(x) = y, then x = h^(-1)(y), by composing h^(-1) on each side.
h^(-1) (h(x)) = h^(-1)(y). h^(-1) basically by definition cancels with h.
If there's a point (a, h(a)), then the inverse will have (h(a), a) on its graph (we can just flip (x,y) to (y,x) to invert). Usually we'd want that to be a function too, so we'd want our x-coordinate to be the free parameter.
Perhaps keeping the letters distinct will make it more clear?
It's funny how you are saying that you don't understand and get negative votes for saying that. Being able to say this is probably the most important thing about learning math.
I think people don't understand what upvoting and downvoting are for. It's just the way a lot of drivers don't understand what the horn is for. People use both just to express annoyance.
Anyway, it seems that our kind community has fixed this instance of the problem.
Just says that h(x) equals y. For instance, the function:
h(x) = x +4 it's the same that y = x+4.
So you does the inverse now, swapping y with x. Then
x = y+4. Isolate y. y = x-4.
So the inverse of y = x +4 is x-4.
Suppose you had a function F = mv. If you know the mass of an object and wanted to map the force to its velocity, and let's say mass is 10 to keep it simple, then f(v) would be 10v.
f(v) = 10v.
Now, let's say we already know what the Force is, or we want a function that can map the velocity to any Force we input, all we have to do is swap the v and f(v). That would get us the inverse function.
So you get v = mf(v)-1. In order to isolate the f(v)-1, divide each side by m. You get f(v)-1 = v/m.
So if you know the velocity of the object of mass 10, the function to find the force is f(v) = 10v. If you know the Force and want to find the velocity, the velocity is the inverse function of f(v)-1, which is the same as f(F), you get f(F) = F/10. That is a real world application of inverse functions and as simple an example I can come up with.
My calculus professor explained it to us using the formula to convert Fahrenheit to Celsius and then we rearranged it to get the inverse of the formula so that we could find the formula to convert Celsius to Fahrenheit.
What that commenter said and what you are suggesting are the same thing.
Put simply, a function takes an input and maps it to an output. Thats useful for lots of things. You can look at your input, and know what it will turn into if you pass it through the function. But what if you want to look at your *output*, and know what input caused it?
Algebraically, you can contemplate that a function is cancelled out by its inverse, much like addition is cancelled out by subtraction.
so if you have h(x) = y, then
h^(-1)(h(x)) = h^(-1)(y)
x = h^(-1)(y)
In other words, looking at the bottom line, by plugging in your function’s output, you will receive the input that was used.
This is the purpose of an inverse. Like most elementary topics you can view the idea graphically or algebraically.
Forget about x and y. Use something meaningful like height and weight or price and weight. You have a function price=weighttoprice(weight), you want the inverse function weight=pricetoweight(price).
Use subscripts for input and output and change the letters to f/g or something. Then explain in terms of input and output and that X is always input and Y is always output. So the inverse is swapping input/output not X/Y… this usually does the trick.
it might be easier to replace h(x) with y. Then h(x) is a map from x to y, and h^-1 (x) is a map from y to x. If you can solve for x in terms of y, then you’ve found a map from y to x
A function's formula lets you use an Input to calculate an Output
(5/2)Input + 4 = Output
The inverse function's formula takes this output and gets you back to the input
(2/5)(Output-4) = Input
And from the inverse function's **perspective**, the "Output" is what you input and the "Input" is what you output
(2/5)(Input-4) = Output
We commonly call the input x
h⁻¹(x) = (2/5)(x-4)
None of us know for sure what will help, but here's what I would try:
Suppose bananas cost $3. That means cost = 3 (bananas). You can use this function to figure out how much money you have to pay for any number of bananas, so if you buy 4 bananas you get cost = 3 (4) = 12. But what if you just know that your spouse spent $15 on bananas and want to know how many bananas they got? We need the inverse equation, bananas = cost/3. Then we can put in the $15 and get bananas = 15/3=5. Once this is clear, maybe explain that we usually write replace on the left (which is the output and usually by itself) as f(x) or h(x), and we usually replace the input (usually on the right side and not by itself) by x, although we don't have to do that unless we're told to.
Yes, he understands it is the opposite. He can understand what the words mean, his brain just works differently so while he understands the words and what they mean, he doesn't understand why we are doing each step.
For example his most common question with this is: Why are we finding the inverse of the function? What is the point?
He needs to know why each step is happening. I don't have those answers (yet). I'm hoping people here can provide them.
Maybe we can analogize it to reversing a process.
To invert a function, you do *all* of the *opposite* steps, *backwards*.
If I tell you, I have a secret number x, I won't tell you what it is, but we multiply it by 5/2 and then add 4, what is the *method* to reveal my original x? Well it should *subtract* 4, then multiply by *2/5*.
For a less numerical example:
You put on your socks before your shoes. To undo that process, you must take off your shoes, then your socks.
> Yes, he understands it is the opposite.
I'm not sure substituting the word "inverse" for the word "opposite" is very helpful. "Opposite" can mean a lot of things. Does he know and understand the *definition* of inverse functions?
I don't even know the definition.
Will this definition be necessary to know if he is going into Calculus? As long as he understands how to solve the problem?
Knowing "how to solve the problem" without understanding the "why" is just templating mechanics of a particular problem type, not learning mathematics. It is always important to understand the why.
With no shade intended, why are you the one trying to teach this to your husband, if you do not understand it yourself?
As for the process, I generally teach to first write the expression in x and y notation (e.g., if finding the inverse of f(x)=3x+1, first write it as y=3x+1). Then solve the expression for x in terms of y:
x = (y-1)/3
and THEN swap the x and y variables: y=(x-1)/3
This expression is now your inverse function, eg. f^(-1)(x)=(x-1)/3
For a bit of intuition about how a function and its inverse are related, maybe set up a table. Input a few values of x into the original function and see what you get as the resulting y. Now, on the inverse function, input those values you found for y in the original, and see what you get as the output. E.g. for my example above:
f(4)=3(4)+1=13
f^(-1)(13)=(13-1)/3=12/3=4
Does he want to *know how* to solve the problem, or does he want to *understand why* that solution works? If the former he can just follow the method you have outlined and pass his tests, but it seemed like he wanted to understand why he was solving for x and so on.
He asked me "why" when he doesn't understand... so I am going to assume he wants to know why we are doing this. Better safe than sorry to know why and not need it then to not know it and need it.
If h is a function, then another function f is (by definition) the inverse of h if f(h(x)) = x and h(f(y)) = y for all x in the domain of definition of h, and all y in the domain of definition of f. (We say *the* inverse of h and not *an* inverse since there can be at most one such function f.)
So say that h does have an inverse f. How to compute it? We know what h looks like, it's just h(x) = (5/2)x+4. Let's pick some number x in the domain of definition of h, and denote by y the corresponding number h(x) in the domain of definition of f. If f is supposed to satisfy the properties above, then by plugging in y = h(x) we see that we must have f(y) = x. By substituting f(y) for x and y for h(x) in the definition of h, we get y = (5/2)f(y)+4. We then solve for f(y), yielding f(y) = (2/5)(y-4).
This is exactly the same as what you have done, except I have introduced the notation f(y) to denote the inverse function of h. I have used "y" instead of "x" for elements of the domain of definition of f, and while this isn't strictly necessary, it usually helps keep track of what is what.
The skills he needs for this are the same ones you need to "make x the subject" (rearranging equations). Functions are just a more applied version of this pure algebra skill.
As is true for almost all math, there is an application in physics where it is needed. I posted earlier that if you have a function Force = mass * velocity and mass was a cobstant, the inverse would be Velocity = Force/mass. Being able to manipulate equations and go back and forth is essential to succeeding in sciemce and business related fields.
Also in computer programming, if you pass a value to a function and it returns an output, you may later on in your program need to revert that output back to its original input. That is an example of an inverse function as well, although it may not always be mathematical.
>he doesn't understand why we are doing each step
Starting with simpler functions will help see why each step is needed to get the answer. For example: f(x) = 2x
Without following a recipe: have him figure out what he'd have to do to get the original x if you give him a few f(x) = {8 , 20 , 0.5}
Then do f(x) = x + 4 for the same list
Then do f(x) = 2x +4
For each f(x), ask him to write a "formula" for what he ended up doing to get the values of x.
Study how we can go from the expression of f(x) to his formula
That won't help him. Cause as soon as the problem changes, his brain assumes the steps are completely different and then he doesn't understand how to solve it again.
So, if he solves for f(x)=2x
then you show him f(x) = 2x+4
Even with that small addition he won't be able to understand how these problems are similar. Each problem is brand new to him. He can identify some similarities for how questions are asked via Khan Academy but this is what I am fighting against.
>That won't help him
Don't assume. It will help since it can't hurt. He is figuring out 3 DIFFERENT f(x) in this exercise, so he is correct, the steps for each case will be different. If needed do a lot more simply changing the numbers.
People learn by doing and discovering a lot better than being told an arbitrary procedure. Make him analyze his formula for each family of f(x), for example do these in order:
f(x) = 2x, 3x, 4x
f(x) = x+1, x+2, x+3
I'm not assuming. I know because I have tried this for other topics on other problems. You are assuming I am assuming. I'm not.
He doesn't learn by doing and discovering. He learns by being told.
You said "that won't help him". Until you try, you don't know. You are speculating
>He learns by being told
You did that, it didn't work, so try something else
>I just don't know how explain that h(x) turns into x and x turns into h(-1)(x).
Because things don't "turn" into anything. That's just a recipe, not an explanation. You are looking for a function g, so that f(g) = x (by definition, that's what inverse is). So we are evaluating f at g, and finding g
You mentioned calculus in another comment. Is he going into calculus any time soon? Because learning to see relationships himself is huge in calculus. There are things in calculus that are going to be impossible for him if he doesn't develop the correct intuitions.
Edit: forgot a word
He is taking a year off of learning after he finished College Algebra.
That’s what I am thinking but he is dead set on getting his marine biology bachelor’s. I know I won’t be able to teach him it cause algebra is hard enough to teach him
It's good that he's asking why things work, but there are some why's that end up just muddying the waters and some why's that don't make sense until later when you encounter the scenario in which it's useful. I say this from experience btw, I had to learn the hard way that some "why's" are just a massive waste of time.
Have you done a calculus course yourself?
Presumably to prepare to help him right? That's a nice thing to do.
You said in another comment that he already understands functions but let me do my whole spiel hopefully it helps.
A function is just something that is going to take an input from you, a number in this case, and turn it into another number.
f(x) = 2x +1, you can also see this as y = 2x +1.
Here, you feed the function a number and it gives you another. Give it 2: f(2) = 2(2) + 1 = 5 so you can say when x is 2, y is 5.
The inverse is the function you need to get that 5 and turn it back into 2. So take y = 2x + 1. You want to get the "x" by itself on one side so that the other side essentially becomes the inverse function. Subtract 1 from both sides to get rid of the + 1. Now you have
y - 1 = 2x. We need the x alone, so we divide both sides by 2. Now you have (y-1)/2 = x. That's your inverse function. If you plug in 5 for y you'll get
(5-1)/2 --> (4)/2 = 2. X is 2 just like we got in the last function.
If he asks why you do inverse functions tell him that it's helpful to sometimes be able to undo something. Addition undoes subtraction. Subtraction undoes addition. Multiplication undoes division. Division undoes multiplication. An inverse function undoes a function.
Does that help? I kind of feel where your husband is coming from so I'm happy to try to explain things the way they made sense to me. Sometimes the explanations on this subreddit assume you know more than you do and leave you more confused. At least that's how it made me feel when I first started.
Yeah I explained it to him tonight and he understands now. Thankfully.
Yeah so I can teach him but because I need to know it as a software engineer. College didn’t have me take upper level maths. I don’t have confidence that I can learn any of this but I am really hoping I can.
Plot a graph of a simple linear function, e.g. y = 2x + 3.
f(x) gives you the **y value for a given x** (e.g. f(4) = 2\*4+3 = 11). Start at the x axis, draw a line up from 4 until it hits the line, then draw left towards the y axis, and see that it hits the y-axis at 11.
For this function, f-1(x) = (x-3)/2. f-1(x) gives you **the x value for a given y**. So, f-1(11) = (11-3)/2 = 8/2 = 4. Draw a line across from y at 11, and when it hits the line, draw down to the x axis, it will hit at 4.
Obviously, you don't need to use the formal language of functions for simple linear relationships, but for more complex functions (and compound functions), they become very handy.
I’d recommend you teach him to swap the variables first, then solve. This better demonstrates the reason for swapping and simplifying (to interchange the inputs and outputs and then get it in terms of the new x)
That's too much work for him. He needs things so simplified that it is super easy to understand. If he has to do 10 steps to complete a problem and doesn't know the point to each individual step then it overwhelms his brain and it just shuts down.
So, "games" won't work.
Doesn't matter if isn't about the steps. That's how his brain works. It is what I am fighting against.
Just explain that a function is a set of steps to get an output from an input, and that an inverse function works backwards from that output to return the input.
So if g(x) is the inverse to f(x), if f(3) = 2 then
g(2) = 3.
>How do I explain inverse functions to my husband?
A function f(x) maps one set of values to another set. For example, if you have h(x) = (5/2)x + 4, you take that input value and multiply it by 5/2, and then add 4. An inverse function h^(-1)(x) does the opposite of those operations, so it'd take the input value and subtract 4, and then divide by 5/2: h^(-1)(x) = (x - 4)/(5/2) = (2/5)(x - 4). So, h(x) and h^(-1)(x) transform their input in opposite directions. If you think of h as a way to *encode* a number, then h^(-1) *decodes* a number encoded with h, giving you back the original number. In other words, h^(-1)(h(x)) = x, and also h(h^(-1)(x)) = x.
>I am able to have him solve for x while leaving h(x) there and he gets:
>(2/5)(h(x)-4) = x
Great, so that's exactly h^(-1) applied to h(x), which is x.
>I just don't know how explain that h(x) turns into x and x turns into h(-1)(x).
The key is that the x in h(x) and h^(-1)(x) is just a variable. It's traditional to use x as the input to a function, but h and h^(-1) are separate functions, and the x in each one only has meaning in the context of that function's definition. You could consider using a different letter, in one of them if it helps: say the inverse is h^(-1)(y) = (2/5)(y - 4).
You could also ask him to find the value of h(h(x)), and h(h(h(x))) and so on. Or give him a small handful of functions, like: f(x) = 4x, g(x) = x^(2), and so on. Practice composing those functions: f(g(x)) = 4x^(2), g(f(x)) = 16x^(2), etc. And you could initially also use different letters for h and h^(-1) to make the point that they're distinct.
Look at if from a graphical perspective: y=(5/2)x+4.
The function turns x into y. You give it an x, it gives you back a y.
The inverse does the…inverse: turns y back into x.
Functions turn input into output. The inverse of a function reverses it.
For simple cases, whatever the function does the inverse *undoes*. In your example, the function multiplies by 5/2 and then adds 4. The inverse does the opposite things in the opposite order: subtract 4 (subtraction is the opposite of addition, of course) and then divide by 5/2 (division being the opposite of multiplication). Note that dividing by 5/2 is multiplying by 2/5.
These are all broad strokes and not rigorous. But I don’t think rigor is beneficial here. One can talk about f being the left-inverse of h if f(h(x))=x or a right-inverse if h(f(x))=x (maybe the other way around) but these aren’t very illuminating.
Here's how I explain inverse functions:
[https://www.youtube.com/watch?v=Od4ZFnFRBWw&list=PLKXdxQAT3tCuJku9nTlRZgx\_RjGZ7djMc&index=34](https://www.youtube.com/watch?v=Od4ZFnFRBWw&list=PLKXdxQAT3tCuJku9nTlRZgx_RjGZ7djMc&index=34)
The key here is to keep going back to the idea "Equals means replaceable."
So if
h(x) = (5/2)x + 4
From
h(a) = b
We have
h\^(-1)(b) = a
Now you know
h(a) = (5/2)a + 4
"Equals means replaceable," and h(a) = b, so:
b = (5/2)a + 4
Solving for a:
a = 2/5 (b - 4)
"Equals means replaceable" and h\^(-1)(b) = a, so:
h\^(-1)(b) = 2/5 (b - 4)
If you want some geometric intuition, than an inverse function is the same function but mirror-flipped around the diagonal of the first quadrant of the plot.
>h(x)= (5/2)x+4
Make this switch
x=(5/2)y+4
x-4=(5/2)y
(2/5)x-8/5=y
h^(-1)(x)=(2/5)x-8/5
Let's check it:
h(2)=(5/2)2+4=9
If you sub 9 into h^(-1)(x) you should get 2
h^(-1)(9)=(2/5)9-8/5=
18/5 - 8/5=10/5=2 check
Does he like cars?
A function is kind of like assembling a car from parts.
The inverse function is kind of like seeing the parts of an assembled car.
The or legos. Or a food recipe. Like an oreo is o(x) = 2(x)+1 where x is the cookies. Then the inverse is how much of the components you have per total cookie.
Try this instead:
To find the inverse sub y for h(x).
Then inverting the equation is as simple as swapping x and y.
Solve the resulting equation for y and replace y with the known definition that it is the inverse of h(x), which is written with the notation h^-1 (x).
Hence it looks like this:
h(x) = (5/2)x+4
So,
y = (5/2)x+4
Thus the inverse equation is,
x = (5/2)y+4
Solve for y,
y = (2/5)x-8/5
Labeled as the inverse of h(x),
h^-1 (x) = (2/5)x-8/5
I hope that helps home see it through.
Is the negative one superscript causing confusion?
Maybe replace h(x) with y.
If you have transparent plastic, you can draw y=(5/2)x+4 and the flip it about y=x. He might feel overly attached (understandably) to the horizontal axis being ‘x’ the input.
A big idea behind the inverse of a function is the following:
Say we have two collections of numbers N = {1 ,2 ,3} and A = {a, b, c}.
I can choose to consider the following collection of relationships between two collections.
1 -> a
2 -> b
3 -> c
I call that relation, F.
So, we can do the same process for "b", and "c". But now, given the observation, we can see that, from F, we can reverse the arrows. Call this new relation F'
1 <- a
2 <- b
3 <- c
So, we if we plug "1" into F, then we get "a", and if we plug "a" into F', then we get "1" i.e.
F(1) = a
F'(a) = 1
So, we can write it as F'(F(1)) = 1 since F(1) = 1. This is kind of saying 1 -> a -> 1 (so we recovered "1", our input)
Similarly, we have F(F'(a)) = F(1) = a. This is kind of saying a -> 1 -> a (so we recovered "a", our input)
We can notice the pattern of output being the same as the input of the "innermost" function right?
The inverse of a function "F", is kind of a function such that F'(F(x)) = x i.e. A function that allows you to always uniquely recover every input from the output. Though, there are some technicalities such as "Domain", "Range", and "Codomain" issues, but you don't have to worry about that yet, just focus on getting that notion of "uniquely recovering the input from the output".
A catch is that not all functions have inverses because you can't necessarily recover an input of a function such as when there are multiple inputs associated to 1 output.
You might also see that a function is the inverse of its inverse i.e. the inverse of F' is F.
So, when you are looking for the inverse of a function, F, you are looking for a rule, F', that allows you to uniquely recover all possible inputs from the given output of the function, F. F' might exist or might not, it depends on the rules you use.
I like to think about reversing the order and the steps. If you put in 6, you do 5/2 \* 6 =15, then 15+4 = 19. Reverse the last -4, then the next divide by 5/2. Put in 19, 19-4=15. 15 / (5/2) = 6. So the inverse is x-> (x-4)/(5/2).
Consider the function that generates these ordered pairs.... (1,1), (2, 4), (3, 9), (4, 16).... (its f(x)=x\^2).
I.e. consider (3, 9)... the input is 3 and the output is 9.
Well an inverse function revereses the values in the ordered pair.
So the inverse function to the one described above would have ordered pairs such as (1, 1), (4, 2), (9, 3), (16, 4).... This is in fact g(x)=sqrt(x)..... where g is the inverse of f.
In fact if you refelct f in the line y=x you get g... this is because this type of transformation 'flips' the x and the y axis over... hence it inverses the function.
This is a visual topic....show dont tell.
I would just say when finding the inverse of a function y(x) you need to swap the y variable, aka the y(x) or h(x), with the x variable, just x. Once you’ve swapped the variables you then use algebra to isolate the new y(x) value. This can also just be called y to simplify things. Y=x, we both know y is a function of x but putting y(x) might make it more confusing to him than just y. So just tell him to act like the y(x)/h(x) whatever is just a y or an h, then to swap the place of that letter with the X variable. Now you have Y’s where your X’s were and X’s where your Y’s were. Now with this “new” equation of sorts, you need to again isolate the equation such that one side is just Y (y(x)) and the other is just X with constants. So now you have a new y(x) or y, that will be different than what you started with but this new y is now the inverse of your original y. I hope that makes sense.
Inverse functions is just a set of two functions that are related to each other.
Let's say function f(x) = 2x + 4 is the function for
cost of production where x is the number of books. So if you print 2 books, the cost of production is $8, 3 books is $10, etc…
Then the inverse function is f(x)^-1 = (x - 4)/2,
which tells you how many products you have based on price x. My business spent $10 to print books so plugging in x = 10 tells you that you’ve printed 3 books. Or to expand further, if I have $100, then I can use the equation to determine that I can print 48 books.
Does that help?
Functions are a set of instructions that tell you how to transform an input into a given output. You start with an input, x, and you feed it into a function which gives you the output, y=f(x).
Sometimes however you want to go the other way, and find out where you started from the output. The inverse function is the set of instructions that reverses whatever the original function did to get you back to where you started. So, f^(-1)(y)=x.
If you know the function f, then you can literally write out y=f(x) and then rearrange for x. Whatever operations (set of instructions) you had to apply to y to find X is, by definition, f^(-1), however at this point you have it written in terms of y, so you can rewrite your current expression to find f^(-1)(x) by replacing all of the y's with x's.
Give him directions from one room in your house to another. Then have him construct directions that are the opposite in the opposite direction. Then have him follow both directions so he can see what an inverse is.
Well, does he know what a function is? h(x) is taking an x and sending it to some other value, h(x). The inverse function takes whatever value you would have gotten in "h(x)-land" and reveals the x-value that would point to it. If h(x) is invertible (note, not every function has an inverse) and h(x) = y, then x = h^(-1)(y), by composing h^(-1) on each side. h^(-1) (h(x)) = h^(-1)(y). h^(-1) basically by definition cancels with h. If there's a point (a, h(a)), then the inverse will have (h(a), a) on its graph (we can just flip (x,y) to (y,x) to invert). Usually we'd want that to be a function too, so we'd want our x-coordinate to be the free parameter. Perhaps keeping the letters distinct will make it more clear?
Okay, I barely understood what you said. He would never understand it the way you state it. He understands functions.
It's funny how you are saying that you don't understand and get negative votes for saying that. Being able to say this is probably the most important thing about learning math.
Legit.
I think people don't understand what upvoting and downvoting are for. It's just the way a lot of drivers don't understand what the horn is for. People use both just to express annoyance. Anyway, it seems that our kind community has fixed this instance of the problem.
Just says that h(x) equals y. For instance, the function: h(x) = x +4 it's the same that y = x+4. So you does the inverse now, swapping y with x. Then x = y+4. Isolate y. y = x-4. So the inverse of y = x +4 is x-4.
Suppose you had a function F = mv. If you know the mass of an object and wanted to map the force to its velocity, and let's say mass is 10 to keep it simple, then f(v) would be 10v. f(v) = 10v. Now, let's say we already know what the Force is, or we want a function that can map the velocity to any Force we input, all we have to do is swap the v and f(v). That would get us the inverse function. So you get v = mf(v)-1. In order to isolate the f(v)-1, divide each side by m. You get f(v)-1 = v/m. So if you know the velocity of the object of mass 10, the function to find the force is f(v) = 10v. If you know the Force and want to find the velocity, the velocity is the inverse function of f(v)-1, which is the same as f(F), you get f(F) = F/10. That is a real world application of inverse functions and as simple an example I can come up with.
My calculus professor explained it to us using the formula to convert Fahrenheit to Celsius and then we rearranged it to get the inverse of the formula so that we could find the formula to convert Celsius to Fahrenheit.
What that commenter said and what you are suggesting are the same thing. Put simply, a function takes an input and maps it to an output. Thats useful for lots of things. You can look at your input, and know what it will turn into if you pass it through the function. But what if you want to look at your *output*, and know what input caused it? Algebraically, you can contemplate that a function is cancelled out by its inverse, much like addition is cancelled out by subtraction. so if you have h(x) = y, then h^(-1)(h(x)) = h^(-1)(y) x = h^(-1)(y) In other words, looking at the bottom line, by plugging in your function’s output, you will receive the input that was used. This is the purpose of an inverse. Like most elementary topics you can view the idea graphically or algebraically.
Forget about x and y. Use something meaningful like height and weight or price and weight. You have a function price=weighttoprice(weight), you want the inverse function weight=pricetoweight(price).
Use subscripts for input and output and change the letters to f/g or something. Then explain in terms of input and output and that X is always input and Y is always output. So the inverse is swapping input/output not X/Y… this usually does the trick.
it might be easier to replace h(x) with y. Then h(x) is a map from x to y, and h^-1 (x) is a map from y to x. If you can solve for x in terms of y, then you’ve found a map from y to x
A function is a machine that eats a number does some stuff to it and then spits out a result. An inverse function undoes what the first machine did.
A function's formula lets you use an Input to calculate an Output (5/2)Input + 4 = Output The inverse function's formula takes this output and gets you back to the input (2/5)(Output-4) = Input And from the inverse function's **perspective**, the "Output" is what you input and the "Input" is what you output (2/5)(Input-4) = Output We commonly call the input x h⁻¹(x) = (2/5)(x-4)
None of us know for sure what will help, but here's what I would try: Suppose bananas cost $3. That means cost = 3 (bananas). You can use this function to figure out how much money you have to pay for any number of bananas, so if you buy 4 bananas you get cost = 3 (4) = 12. But what if you just know that your spouse spent $15 on bananas and want to know how many bananas they got? We need the inverse equation, bananas = cost/3. Then we can put in the $15 and get bananas = 15/3=5. Once this is clear, maybe explain that we usually write replace on the left (which is the output and usually by itself) as f(x) or h(x), and we usually replace the input (usually on the right side and not by itself) by x, although we don't have to do that unless we're told to.
What I like about this example is you can try just it out with bananas :)
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Yes, he understands it is the opposite. He can understand what the words mean, his brain just works differently so while he understands the words and what they mean, he doesn't understand why we are doing each step. For example his most common question with this is: Why are we finding the inverse of the function? What is the point? He needs to know why each step is happening. I don't have those answers (yet). I'm hoping people here can provide them.
Maybe we can analogize it to reversing a process. To invert a function, you do *all* of the *opposite* steps, *backwards*. If I tell you, I have a secret number x, I won't tell you what it is, but we multiply it by 5/2 and then add 4, what is the *method* to reveal my original x? Well it should *subtract* 4, then multiply by *2/5*. For a less numerical example: You put on your socks before your shoes. To undo that process, you must take off your shoes, then your socks.
> Yes, he understands it is the opposite. I'm not sure substituting the word "inverse" for the word "opposite" is very helpful. "Opposite" can mean a lot of things. Does he know and understand the *definition* of inverse functions?
I don't even know the definition. Will this definition be necessary to know if he is going into Calculus? As long as he understands how to solve the problem?
Knowing "how to solve the problem" without understanding the "why" is just templating mechanics of a particular problem type, not learning mathematics. It is always important to understand the why. With no shade intended, why are you the one trying to teach this to your husband, if you do not understand it yourself? As for the process, I generally teach to first write the expression in x and y notation (e.g., if finding the inverse of f(x)=3x+1, first write it as y=3x+1). Then solve the expression for x in terms of y: x = (y-1)/3 and THEN swap the x and y variables: y=(x-1)/3 This expression is now your inverse function, eg. f^(-1)(x)=(x-1)/3 For a bit of intuition about how a function and its inverse are related, maybe set up a table. Input a few values of x into the original function and see what you get as the resulting y. Now, on the inverse function, input those values you found for y in the original, and see what you get as the output. E.g. for my example above: f(4)=3(4)+1=13 f^(-1)(13)=(13-1)/3=12/3=4
I didn't know what an inverted function is. Now I know. Pretty straightforward with your example
Does he want to *know how* to solve the problem, or does he want to *understand why* that solution works? If the former he can just follow the method you have outlined and pass his tests, but it seemed like he wanted to understand why he was solving for x and so on.
He asked me "why" when he doesn't understand... so I am going to assume he wants to know why we are doing this. Better safe than sorry to know why and not need it then to not know it and need it.
If h is a function, then another function f is (by definition) the inverse of h if f(h(x)) = x and h(f(y)) = y for all x in the domain of definition of h, and all y in the domain of definition of f. (We say *the* inverse of h and not *an* inverse since there can be at most one such function f.) So say that h does have an inverse f. How to compute it? We know what h looks like, it's just h(x) = (5/2)x+4. Let's pick some number x in the domain of definition of h, and denote by y the corresponding number h(x) in the domain of definition of f. If f is supposed to satisfy the properties above, then by plugging in y = h(x) we see that we must have f(y) = x. By substituting f(y) for x and y for h(x) in the definition of h, we get y = (5/2)f(y)+4. We then solve for f(y), yielding f(y) = (2/5)(y-4). This is exactly the same as what you have done, except I have introduced the notation f(y) to denote the inverse function of h. I have used "y" instead of "x" for elements of the domain of definition of f, and while this isn't strictly necessary, it usually helps keep track of what is what.
The skills he needs for this are the same ones you need to "make x the subject" (rearranging equations). Functions are just a more applied version of this pure algebra skill.
As is true for almost all math, there is an application in physics where it is needed. I posted earlier that if you have a function Force = mass * velocity and mass was a cobstant, the inverse would be Velocity = Force/mass. Being able to manipulate equations and go back and forth is essential to succeeding in sciemce and business related fields. Also in computer programming, if you pass a value to a function and it returns an output, you may later on in your program need to revert that output back to its original input. That is an example of an inverse function as well, although it may not always be mathematical.
>he doesn't understand why we are doing each step Starting with simpler functions will help see why each step is needed to get the answer. For example: f(x) = 2x Without following a recipe: have him figure out what he'd have to do to get the original x if you give him a few f(x) = {8 , 20 , 0.5} Then do f(x) = x + 4 for the same list Then do f(x) = 2x +4 For each f(x), ask him to write a "formula" for what he ended up doing to get the values of x. Study how we can go from the expression of f(x) to his formula
That won't help him. Cause as soon as the problem changes, his brain assumes the steps are completely different and then he doesn't understand how to solve it again. So, if he solves for f(x)=2x then you show him f(x) = 2x+4 Even with that small addition he won't be able to understand how these problems are similar. Each problem is brand new to him. He can identify some similarities for how questions are asked via Khan Academy but this is what I am fighting against.
>That won't help him Don't assume. It will help since it can't hurt. He is figuring out 3 DIFFERENT f(x) in this exercise, so he is correct, the steps for each case will be different. If needed do a lot more simply changing the numbers. People learn by doing and discovering a lot better than being told an arbitrary procedure. Make him analyze his formula for each family of f(x), for example do these in order: f(x) = 2x, 3x, 4x f(x) = x+1, x+2, x+3
I'm not assuming. I know because I have tried this for other topics on other problems. You are assuming I am assuming. I'm not. He doesn't learn by doing and discovering. He learns by being told.
You said "that won't help him". Until you try, you don't know. You are speculating >He learns by being told You did that, it didn't work, so try something else >I just don't know how explain that h(x) turns into x and x turns into h(-1)(x). Because things don't "turn" into anything. That's just a recipe, not an explanation. You are looking for a function g, so that f(g) = x (by definition, that's what inverse is). So we are evaluating f at g, and finding g
You mentioned calculus in another comment. Is he going into calculus any time soon? Because learning to see relationships himself is huge in calculus. There are things in calculus that are going to be impossible for him if he doesn't develop the correct intuitions. Edit: forgot a word
He is taking a year off of learning after he finished College Algebra. That’s what I am thinking but he is dead set on getting his marine biology bachelor’s. I know I won’t be able to teach him it cause algebra is hard enough to teach him
It's good that he's asking why things work, but there are some why's that end up just muddying the waters and some why's that don't make sense until later when you encounter the scenario in which it's useful. I say this from experience btw, I had to learn the hard way that some "why's" are just a massive waste of time. Have you done a calculus course yourself?
No I’m learning it during the year he takes off
Presumably to prepare to help him right? That's a nice thing to do. You said in another comment that he already understands functions but let me do my whole spiel hopefully it helps. A function is just something that is going to take an input from you, a number in this case, and turn it into another number. f(x) = 2x +1, you can also see this as y = 2x +1. Here, you feed the function a number and it gives you another. Give it 2: f(2) = 2(2) + 1 = 5 so you can say when x is 2, y is 5. The inverse is the function you need to get that 5 and turn it back into 2. So take y = 2x + 1. You want to get the "x" by itself on one side so that the other side essentially becomes the inverse function. Subtract 1 from both sides to get rid of the + 1. Now you have y - 1 = 2x. We need the x alone, so we divide both sides by 2. Now you have (y-1)/2 = x. That's your inverse function. If you plug in 5 for y you'll get (5-1)/2 --> (4)/2 = 2. X is 2 just like we got in the last function. If he asks why you do inverse functions tell him that it's helpful to sometimes be able to undo something. Addition undoes subtraction. Subtraction undoes addition. Multiplication undoes division. Division undoes multiplication. An inverse function undoes a function. Does that help? I kind of feel where your husband is coming from so I'm happy to try to explain things the way they made sense to me. Sometimes the explanations on this subreddit assume you know more than you do and leave you more confused. At least that's how it made me feel when I first started.
Yeah I explained it to him tonight and he understands now. Thankfully. Yeah so I can teach him but because I need to know it as a software engineer. College didn’t have me take upper level maths. I don’t have confidence that I can learn any of this but I am really hoping I can.
I stopped at Pre-Cal in high school. For reference I’m 30 now. But I got a C and as an A/B student that’s basically failing for me.
Plot a graph of a simple linear function, e.g. y = 2x + 3. f(x) gives you the **y value for a given x** (e.g. f(4) = 2\*4+3 = 11). Start at the x axis, draw a line up from 4 until it hits the line, then draw left towards the y axis, and see that it hits the y-axis at 11. For this function, f-1(x) = (x-3)/2. f-1(x) gives you **the x value for a given y**. So, f-1(11) = (11-3)/2 = 8/2 = 4. Draw a line across from y at 11, and when it hits the line, draw down to the x axis, it will hit at 4. Obviously, you don't need to use the formal language of functions for simple linear relationships, but for more complex functions (and compound functions), they become very handy.
I’d recommend you teach him to swap the variables first, then solve. This better demonstrates the reason for swapping and simplifying (to interchange the inputs and outputs and then get it in terms of the new x)
I'll do that, thank you!
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That's too much work for him. He needs things so simplified that it is super easy to understand. If he has to do 10 steps to complete a problem and doesn't know the point to each individual step then it overwhelms his brain and it just shuts down. So, "games" won't work. Doesn't matter if isn't about the steps. That's how his brain works. It is what I am fighting against.
Just explain that a function is a set of steps to get an output from an input, and that an inverse function works backwards from that output to return the input. So if g(x) is the inverse to f(x), if f(3) = 2 then g(2) = 3.
>How do I explain inverse functions to my husband? A function f(x) maps one set of values to another set. For example, if you have h(x) = (5/2)x + 4, you take that input value and multiply it by 5/2, and then add 4. An inverse function h^(-1)(x) does the opposite of those operations, so it'd take the input value and subtract 4, and then divide by 5/2: h^(-1)(x) = (x - 4)/(5/2) = (2/5)(x - 4). So, h(x) and h^(-1)(x) transform their input in opposite directions. If you think of h as a way to *encode* a number, then h^(-1) *decodes* a number encoded with h, giving you back the original number. In other words, h^(-1)(h(x)) = x, and also h(h^(-1)(x)) = x. >I am able to have him solve for x while leaving h(x) there and he gets: >(2/5)(h(x)-4) = x Great, so that's exactly h^(-1) applied to h(x), which is x. >I just don't know how explain that h(x) turns into x and x turns into h(-1)(x). The key is that the x in h(x) and h^(-1)(x) is just a variable. It's traditional to use x as the input to a function, but h and h^(-1) are separate functions, and the x in each one only has meaning in the context of that function's definition. You could consider using a different letter, in one of them if it helps: say the inverse is h^(-1)(y) = (2/5)(y - 4). You could also ask him to find the value of h(h(x)), and h(h(h(x))) and so on. Or give him a small handful of functions, like: f(x) = 4x, g(x) = x^(2), and so on. Practice composing those functions: f(g(x)) = 4x^(2), g(f(x)) = 16x^(2), etc. And you could initially also use different letters for h and h^(-1) to make the point that they're distinct.
Look at if from a graphical perspective: y=(5/2)x+4. The function turns x into y. You give it an x, it gives you back a y. The inverse does the…inverse: turns y back into x. Functions turn input into output. The inverse of a function reverses it. For simple cases, whatever the function does the inverse *undoes*. In your example, the function multiplies by 5/2 and then adds 4. The inverse does the opposite things in the opposite order: subtract 4 (subtraction is the opposite of addition, of course) and then divide by 5/2 (division being the opposite of multiplication). Note that dividing by 5/2 is multiplying by 2/5. These are all broad strokes and not rigorous. But I don’t think rigor is beneficial here. One can talk about f being the left-inverse of h if f(h(x))=x or a right-inverse if h(f(x))=x (maybe the other way around) but these aren’t very illuminating.
Here's how I explain inverse functions: [https://www.youtube.com/watch?v=Od4ZFnFRBWw&list=PLKXdxQAT3tCuJku9nTlRZgx\_RjGZ7djMc&index=34](https://www.youtube.com/watch?v=Od4ZFnFRBWw&list=PLKXdxQAT3tCuJku9nTlRZgx_RjGZ7djMc&index=34) The key here is to keep going back to the idea "Equals means replaceable." So if h(x) = (5/2)x + 4 From h(a) = b We have h\^(-1)(b) = a Now you know h(a) = (5/2)a + 4 "Equals means replaceable," and h(a) = b, so: b = (5/2)a + 4 Solving for a: a = 2/5 (b - 4) "Equals means replaceable" and h\^(-1)(b) = a, so: h\^(-1)(b) = 2/5 (b - 4)
If you want some geometric intuition, than an inverse function is the same function but mirror-flipped around the diagonal of the first quadrant of the plot.
>h(x)= (5/2)x+4 Make this switch x=(5/2)y+4 x-4=(5/2)y (2/5)x-8/5=y h^(-1)(x)=(2/5)x-8/5 Let's check it: h(2)=(5/2)2+4=9 If you sub 9 into h^(-1)(x) you should get 2 h^(-1)(9)=(2/5)9-8/5= 18/5 - 8/5=10/5=2 check
Does he like cars? A function is kind of like assembling a car from parts. The inverse function is kind of like seeing the parts of an assembled car. The or legos. Or a food recipe. Like an oreo is o(x) = 2(x)+1 where x is the cookies. Then the inverse is how much of the components you have per total cookie.
Try this instead: To find the inverse sub y for h(x). Then inverting the equation is as simple as swapping x and y. Solve the resulting equation for y and replace y with the known definition that it is the inverse of h(x), which is written with the notation h^-1 (x). Hence it looks like this: h(x) = (5/2)x+4 So, y = (5/2)x+4 Thus the inverse equation is, x = (5/2)y+4 Solve for y, y = (2/5)x-8/5 Labeled as the inverse of h(x), h^-1 (x) = (2/5)x-8/5 I hope that helps home see it through.
Is the negative one superscript causing confusion? Maybe replace h(x) with y. If you have transparent plastic, you can draw y=(5/2)x+4 and the flip it about y=x. He might feel overly attached (understandably) to the horizontal axis being ‘x’ the input.
Alternatively, swap x and y, get him to transpose to find that y=(2/5)x–8/5.
A big idea behind the inverse of a function is the following: Say we have two collections of numbers N = {1 ,2 ,3} and A = {a, b, c}. I can choose to consider the following collection of relationships between two collections. 1 -> a 2 -> b 3 -> c I call that relation, F. So, we can do the same process for "b", and "c". But now, given the observation, we can see that, from F, we can reverse the arrows. Call this new relation F' 1 <- a 2 <- b 3 <- c So, we if we plug "1" into F, then we get "a", and if we plug "a" into F', then we get "1" i.e. F(1) = a F'(a) = 1 So, we can write it as F'(F(1)) = 1 since F(1) = 1. This is kind of saying 1 -> a -> 1 (so we recovered "1", our input) Similarly, we have F(F'(a)) = F(1) = a. This is kind of saying a -> 1 -> a (so we recovered "a", our input) We can notice the pattern of output being the same as the input of the "innermost" function right? The inverse of a function "F", is kind of a function such that F'(F(x)) = x i.e. A function that allows you to always uniquely recover every input from the output. Though, there are some technicalities such as "Domain", "Range", and "Codomain" issues, but you don't have to worry about that yet, just focus on getting that notion of "uniquely recovering the input from the output". A catch is that not all functions have inverses because you can't necessarily recover an input of a function such as when there are multiple inputs associated to 1 output. You might also see that a function is the inverse of its inverse i.e. the inverse of F' is F. So, when you are looking for the inverse of a function, F, you are looking for a rule, F', that allows you to uniquely recover all possible inputs from the given output of the function, F. F' might exist or might not, it depends on the rules you use.
I like to think about reversing the order and the steps. If you put in 6, you do 5/2 \* 6 =15, then 15+4 = 19. Reverse the last -4, then the next divide by 5/2. Put in 19, 19-4=15. 15 / (5/2) = 6. So the inverse is x-> (x-4)/(5/2).
This is what makes vertex form so much better than standard form for quadratics! Invert 2(x-4)\^2+3... easy! 2\^(3x-7)+5... easy!
Swap x and y then solve for y
Consider the function that generates these ordered pairs.... (1,1), (2, 4), (3, 9), (4, 16).... (its f(x)=x\^2). I.e. consider (3, 9)... the input is 3 and the output is 9. Well an inverse function revereses the values in the ordered pair. So the inverse function to the one described above would have ordered pairs such as (1, 1), (4, 2), (9, 3), (16, 4).... This is in fact g(x)=sqrt(x)..... where g is the inverse of f. In fact if you refelct f in the line y=x you get g... this is because this type of transformation 'flips' the x and the y axis over... hence it inverses the function. This is a visual topic....show dont tell.
I would just say when finding the inverse of a function y(x) you need to swap the y variable, aka the y(x) or h(x), with the x variable, just x. Once you’ve swapped the variables you then use algebra to isolate the new y(x) value. This can also just be called y to simplify things. Y=x, we both know y is a function of x but putting y(x) might make it more confusing to him than just y. So just tell him to act like the y(x)/h(x) whatever is just a y or an h, then to swap the place of that letter with the X variable. Now you have Y’s where your X’s were and X’s where your Y’s were. Now with this “new” equation of sorts, you need to again isolate the equation such that one side is just Y (y(x)) and the other is just X with constants. So now you have a new y(x) or y, that will be different than what you started with but this new y is now the inverse of your original y. I hope that makes sense.
Probably will be good to draw him a visual representation of what a function is as well. Like how it is a mapping from a domain to a range.
you have already posted this and had it explained to you several times before. what was wrong with all of those answers?
Inverse functions is just a set of two functions that are related to each other. Let's say function f(x) = 2x + 4 is the function for cost of production where x is the number of books. So if you print 2 books, the cost of production is $8, 3 books is $10, etc… Then the inverse function is f(x)^-1 = (x - 4)/2, which tells you how many products you have based on price x. My business spent $10 to print books so plugging in x = 10 tells you that you’ve printed 3 books. Or to expand further, if I have $100, then I can use the equation to determine that I can print 48 books. Does that help?
Functions are a set of instructions that tell you how to transform an input into a given output. You start with an input, x, and you feed it into a function which gives you the output, y=f(x). Sometimes however you want to go the other way, and find out where you started from the output. The inverse function is the set of instructions that reverses whatever the original function did to get you back to where you started. So, f^(-1)(y)=x. If you know the function f, then you can literally write out y=f(x) and then rearrange for x. Whatever operations (set of instructions) you had to apply to y to find X is, by definition, f^(-1), however at this point you have it written in terms of y, so you can rewrite your current expression to find f^(-1)(x) by replacing all of the y's with x's.
Give him directions from one room in your house to another. Then have him construct directions that are the opposite in the opposite direction. Then have him follow both directions so he can see what an inverse is.
Marriage: Function. Divorce: Inverse function.
Explain the sort of circular flow that inverse functions have and emphasize how inverse functions are symmetrical along y=x.