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Inverse Functions

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Today we're going to talk about inverse functions. So we'll start by talking about what a inverse function is, and then we'll do some examples looking at inverse functions, both algebraically and graphically. So let's start by talking about inverse functions and the notation that we use.

Suppose I have some function f, and I take an input value of x, and I use that for my function, f. So my function f is some combination of one or more operations that I would do to my input, x. And the result is going to be my output value, f of x. And we also say that f of x, or the output, is equal to variable y. So you can use y, or f of x interchangeably.

So now let's say I take my output value, and I use that as the input for another function, the inverse of my original function. So whatever the operations were for my original function, f, I'm going to use the inverse operations here. So this is the inverse of my original f function.

So if I use my output as the input from my inverse function, then my output now is going to be equal to x, which is the original input of my first function, f. And that makes sense, because if I take in my input, I do some operations, and then I take what I get and I do the opposite of that, I should end up back where I started.

Now we can also write this as f inverse of f of x is equal to x, again, because y then f of x are the same thing. And this again just says that if I take the inverse function of an original function, f of x, I'll come back to my original input value.

So let's look at an example of inverse functions. Let's say I have a function, f of x. And that function undergoes two operations. We multiply by 3, and we add 2. So algebraically we can write f of x as 3x plus 2. So for example, if I wanted to use an input of 4-- so I'm going to use 4 for my input value for x. And if I wanted to find f of 4, I would input 4 for my x here, and then simplify. So 3 times 4 will give me 12 . And 12 plus 2 will give me 14. So when my input is 4 for my f of x, my output is 14.

Now if we wanted to find the inverse of of my function f of x, the inverse of f of x is going to be using the inverse operations in the opposite order. So we're going to use the inverse operations of adding 2 and multiplying by 3 in opposite order. So the inverse of adding 2 is subtracting 2, and the inverse of multiplying by 3 is dividing by 3. So we can define f inverse of x algebraically as x minus 2, subtracting 2, divided by 3.

So if I use my output value of 14 as my input value for my inverse function, we should have a output value of 4, our original input value. So let's try that. If I use f inverse, or if I find f inverse of 14, I'm going to substitute 14 here for x. Simplifying, 14 minus 2 is 12, and 12 over 3 is 4. So I found my original input value, verifying that these two are inverse functions of each other.

So let's look at a graph of a function, and the graph of its inverse. I've got the function f of x, which is equal to 2 x minus 4, and its graphed here in red. And I've got the inverse of that function, f inverse of x, which would be equal to x plus 4 over 2. And it's graphed here, in green.

Now inverse functions have the property that they are symmetric about the line, y equals x. So the line, y equals x, is a diagonal line through the coordinate plane, and it crosses every point where x and y have the same value. So it will cross the point 1,1 and 2,2, and 3, 3. So it creates a perfectly 45 degree angle with the x-axis.

And so our graphs for f of x and f inverse of x are symmetric around this line, y equal x. So we can see that by looking at some point on our line. So first, if we look at a point on this line 2x minus 4-- let's day the point here, which is 1 negative 2. My x value is one, and my y value is negative 2. If I were to find this corresponding point on the point that's symmetric to it on my line for the inverse function, I would see that that is at the point negative 2, 1.

So I can see that the x value became my y value, and my y value became my x value. Let's check another point. This point here on my line for 2x minus 4, is the 0.32. If I look for the point that's symmetric to that on my inverse function line, I see that that is at the point 2, 3. So again, my x value became y, and my y value became my x value.

So let's over our key points from today. In a function, f of x, an input value undergoes one or more operations, resulting in an output value. If the resulting output value is the argument of the functions inverse, it will return to the original input value. The function, f of x, describes a set of operations done to an input, x. The inverse function, f inverse of x, describes the inverse operations of the original function f of x. And the line, y equals x, is a line of symmetry, of the function, and its inverse.

So I hope that these key points and examples helped you understand a little bit more about inverse functions. Keep using your notes, and keep on practicing, and soon you'll be a pro. Thanks for watching.