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Finding the Inverse of a Function

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Today we're going to talk about finding the inverse of a function. So we'll start by reviewing what the inverse of a function is, and then we'll do some examples, finding the inverse graphically and algebraically.

So let's start by reviewing what an inverse function is. An inverse function are basically the operations that undo the operations of your original function. So for example, if I have a function f, and I input x. So x is my input, and I do some operations to it. That's my f function. My output is going to be y. So x is my input, y is my output.

Now if I take my output, y, and I use that as the input for the inverse function, f, so f inverse, f inverse of y is going to bring me back to my original input of x. And that's again because the inverse function undoes the operations of the x function.

So if I started with x, I do the operations of the x function, I get y. Then if I start with y, and I do the inverse operations, then I'll come back to my original value of x. We can also write here, f inverse of f of x, because y is the same as f of x. So f inverse of f of x is just going to give me x.

So let's start by talking about how to find the inverse of a function graphically. I've got the function f of x is equal to 2x minus 4. It's inverse, f inverse of x, is equal to one half x plus 2. So we know that all points on the function f of x can be described as xy, and then all points on the inverse of that function, f inverse of x, would be described as yx, where x and y are the coordinates of the original function.

So for example, if I have an input value of x equal to 1, the output, the y value, or f of x, is going to be negative 2. So I have the point 1, negative 2 for my original function f of x. But if I take an input value of negative 2, then the output f inverse of x, or the y value, is going to be 1 on my inverse function graph. So on my inverse function graph I have the point negative 2, 1-- negative 2, positive 1. So we can see that the value of x equals to one for our function f of x is the y value of our inverse function for the input value of x equals negative 2.

So now let's talk about how to find the inverse of a function algebraically. I've got the function y equals negative 3 x plus 12. To find the inverse of this function, we can start by swapping our x and y variables, and then solving the equation for y. An alternative method would be to write the equation as x is equal to some expression, and then swapping the x and y variables.

So I'm going to start by rewriting this equation. I'm going to swap my x and y variables, and write this as x is equal to negative 3y plus 12. Now I'll solve this for y by isolating the y variable. So I'm going to subtract 12 from both sides. So this becomes x minus 12 is equal to negative 3y. Then I'll divide both sides by negative 3. Here this will cancel, so I just have y equals. And on this side when I divide, I'm going to rewrite my fraction. I'm going to separate it into two fractions. So the first fraction will be x over negative 3, which is the same as negative 1 over 3 x. And then I have negative 12 over negative 3, which would give me a positive 4.

So I found that my inverse function is y equals negative 1/3 x plus 4. So if we call this function-- call this f of x, we say this is equal to f of x, then this is going to be equal to f inverse of x. And as we talked about before, the operations of a function and its inverse undo each other. So we know that if we find the inverse of our original function, f of x, then it will just be equal to x.

So we can show that with this example by finding the inverse of our original function, f of x. So if I want to find the inverse of my original function, f of x, then I'm going to use my expression for f of x as the input for my inverse function. So I'm going to start writing this as negative 1 over 3 times. Instead of x, I'm going to write this expression, negative 3 x plus 12, and then I will have the plus 4 at the end.

So now when I simplify this, I'm going to start by multiplying negative 1/3 times a negative 3. It's just going to give me 1. So this will be 1x, or just x. And then negative 1/3 times a positive 12 will give me a minus 4. And if I bring down my positive 4 at the end, I see that these two terms will cancel out. So I'm just left with x. So I found that f inverse of f of x is equal to x.

So let's go over our key points from today. The inverse of a function undoes the operations of the function. In other words, f inverse of f of x equals x. All points on the curve of f of x can be described as xy. All points on the curve f inverse of x can be described as yx, where x and y are the coordinates of the original function. And the inverse can be found algebraically by swapping x and y, and then solving the equation for y.

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