Understanding and Graphing the Inverse Function

If I tell you that I have a function that maps the number of feet in some distance to the number of inches in that distance, you might tell me that the function would be written as y = f(x) where x is the number of feet and that’s the inputthe function.

and y is the number of inches and that’s the output of the function.

you might even tell me that if that is my function that takes the number of feet and gives you the number of inches, that function f(x)= 12 x

because for every foot that you have, every x that you have, you have 12 inches in that foot.

But what if I told you that I wanted a function that does the exact opposite?

I want it to take the number of inches in some distance, and return the number of feet in that distance.

Could you tell me what this function is?

well Inverse functions are exactly that.

If we have a function like y = f(x) , then the inverse function which we were write as y = f -1( x ), does the exact opposite of the function.

so what happens if you put a function and its inverse into a composite function.

so first, we evaluate this inner function, then we’re gonna evaluate the outer function.

so Let’s take a look at an example.

Say we start with 4 feet.

Well, our function is 12 x because there are 12 inches in every foot.

and If we plug 4 in to our function, we end up with f(x) were x is 4 is 48 inches.

Now if we take the inverse function, and the inverse function, we have to be f -1 ( x ) = x (1/12).

So, if we take 48 inches, that gives us back, 4 feet.

Okay, so you might be able to find f(x) and f -1( x ) just based on your knowledge of inches and feet, but how do you do it in general?

first you wanna Write out a function in terms of x and y, like y = f(x) .

then the way we’re gonna do it is we’re going to Swap the x and y variables.

we’re gonna Solve for y as a function of x .

and we’re gonna take the result and call that the inverse function.

now we’re gonna Check our answers.

so let’s do this

let’s say we have a function, f(x) = 3( x - 1) + 2.

We’re gonna write this out in terms of x and y : y = 3( x - 1) + 2.

Then we’re gonna swap the x and y variables, ok x = 3( y - 1) + 2.

This can be a kind of confusing step if you’re not careful, but at its hard, all you’re doing is putting x everywhere you see y and putting y everywhere you see x .

Then you’re gonna solve for y as a function of x.

So I’m going to subtract 2 from both sides, i’m gonna devide both sides by 3, now i’m going to add 1 to both sides and I end up with y = 1 + ( x - 2)/3.

I’m gonna call what’s on the right hand side here, the inverse function of x

so my variable is still x, but now its the inverse function.

Finally, I’m going to check my answer, so I’m going to find f -1 of ( f(x) ).

so to do this, I’m going to write f(x) = 3( x -1) + 2.

I’m going to plug in my f(x) as input to my f inverse of x.

that means i’m gonna plug this in, right here

that is now my input.

so now I have my input here, and I’m just going to solve, i’m gonna simplify.

And sure enough, the inverse function of f(x) gives me back x , which is exactly what we’d expect.

So what about a function like y = round( x )?

so remember that round( x ) just rounds our input to the nearest integer, so that gives me 4.

However, round(4.8) give me 5 and say round(5.1) will also give me 5.

so In this case, do you ever think that you can find a function that will take 5 and give your either 5.1 or 4.8?

well No, round( x ) has no inverse.

so What about the function like f(x) = x 3 + 3 x ?

so I can write it out in terms of x and y

I can swap the variables,

I can then solve it for y, but i don’t know how to really do that.

that’s not obvious to me.

so Is there another way?

Let’s go back and look at something like, 3 x - 6.

so an easier function.

I end up with a graph that looks like this, it’s a simple line.

Now I’m going to graph the inverse, which is ( x + 6)/3.

what happened if i graph the inverse?

the is this blue line; so the inverse looks a lot like the original function, except it’s mirrored.

And it’s actually mirrored over the 45-degree angle, which is the x = y line.

so If I could fold this paper in half, I’d see that the function and its inverse become the same line.

so I can use this on much more complex functions.

Say I have some function f(x) that looks like that.

If I draw the 45-degree line and i mirror it, i get a pretty good idea of what that inverse function looks like.

so the recap.

The inverse function will undo the function.

That means that the inverse function of the function will give you back what you started with.

But not all functions will have inverses.

For example, round didn’t have an inverse.

You can find the inverse function with our five-step process.

and If you graph the function and its inverse, they’re 45-degree reflections of one another.

so that’s an easy way to find the inverse or get an ideas to what the inverse function looks like for really complex functions.