Hi, and welcome. My name is Anthony Varela. And today we're going to talk about inverse functions.
So we'll start off by defining an inverse function and talk about its notation. We are going to write a function's inverse. And then we'll also take a look at the graph of a function and its inverse.
So first, what is an inverse function? When thinking about a function we have a value. We can call it x. And it goes into this function where operations are applied to it, so there's some output value f of x We can also call this a y value.
Now, what would happen then if we take this output-- so f of x-- and put it into the function's inverse? So this is going to undo everything that's our x values went through in the function. So the result then is what we started with, x.
So the inverse function, it undoes the operations of a function. And this is how we write it. We use this negative 1, and it doesn't mean that we're raising the function to the negative first power. That's just how we indicate that we're talking about a function's inverse.
So there is this relationship between a function and its inverse. I could say that if the input of an inverse function is the function f of x, the result then is just going to be what we started with, x. So we'll come back to this later.
Next, I'd like to talk about inverse operations and inverse functions. So first we're going to design a function, and then we'll find its inverse. So we're starting with a value x. And we'll add 5 to that value. So far, our function is x plus 5. And then we'll double this quantity. So our function f of x is 2 times x plus 5.
We're going to use this function f of x to write the inverse function. And the big idea here is that we're using inverse operations. And we're going to apply them in reverse order.
So to start off our inverse function, we're going to start with a value, our argument, x. And we need to undo our step of doubling our quantity up here. So the opposite of doubling is to divide by 2. So I'm going to divide x by 2.
Next, we have to undo adding 5. So that would be subtracting 5. So I'm subtracting 5 from x/2. And this is our inverse function. So I applied inverse operations in the reverse order.
So we have established then that f of x equals 2 times x plus 5. And the inverse function is x/2 minus 5. And earlier, I said there's this relationship that the inverse of a function will give us the original argument. So let's go ahead and prove this.
So f of x is 2 times x plus 5. So I put that in here as the argument to the inverse function. So now what I'm going to do is take a look at my expression for the inverse.
And instead of writing x, I'm going to write 2 times x plus 5. So here is 2 times x plus 5 over 2 minus 5. That's what I have here. And I have to show that this equals x.
Well, notice I have a 2 in both the numerator and denominator here, so that cancels. So I have x plus 5 minus 5. Well, my plus 5 and my minus 5 cancel, so I have that x equals x. So I've shown that with my function f of x and my inverse function that this property always holds true.
Lastly, we're going to talk about a function and its inverse on a graph. So here I have graphed our function 2 times x plus 5. That's the blue line right here. And it's inverse-- this is x/2 minus 5-- this is the red line right here.
And a characteristic of a function and its inverse on a graph is that there is a line of reflection. And this line of reflection is always y equals x. So there's this line of reflection between a function and its inverse. It is the line y equals x.
So let's take a look at a couple of coordinate points on our function and its inverse. So here is the point negative 3, 4. And if we reflect this point across y equals x, its corresponding point is 4, negative 3. Well, here's a point on the red line-- 2, negative 4. And if we reflect this across the line y equals x, we get negative 4 comma 2.
Now, did you notice anything when we were reflecting our points? Well, we would just take our coordinates x, y, and on the inverse function, we just swap x and y. And those coordinates are y comma x. So there is another interesting characteristic of the coordinates of points on a function and its inverse. You can just swap x and y and plot them on the coordinate plane.
So let's review our lesson on inverse functions. Well, the inverse function undoes operations of a function. So if you have a function f of x, you can apply inverse operations in reverse order to write the inverse function.
And on a graph, the line y equals x represents a line of reflection between a function and its inverse. So you can take the coordinate points x, y, just swap y and x, and you will have coordinates for your inverse function.
So thanks for watching this tutorial on inverse functions. Hope to see you next time.