Hi and welcome. My name is Anthony Varela. And today, we're going to talk about different ways to find the inverse of a function. So we'll review the relationship between a function and its inverse. We'll look at how to evaluate an inverse function using a graph, and then we'll practice writing an inverse function algebraically.
So first, let's review this relationship between a function and its inverse. So if we have a function f of x, we're taking x. And we're applying some number of operations to x. And we get an output value. We can call that y. Well, if we use this output value y as the input to the functions inverse, it's going to undo everything that the function did. And we'll get back our original input x.
So there's this relationship then between a function and its inverse that if the function is used as the input to its inverse, that will equal x. So that's the relationship then between a function and its inverse. It undoes the operations of the function.
So we're going to use a graph of a function to evaluate its inverse. So here, we have the graph of y equals f of x. And we would like to find the inverse function at 1. So the interesting thing about the graph of a function and its inverse is that all points on the function can be represented by the coordinates x, y. And all points on the inverse would be represented by the coordinates y, x. So x and y have switched going from the function to its inverse.
So to find this inverse at x equals 1, we're actually going to locate 1 on the y-axis, not the x-axis. And then we'll find the corresponding x value. So we're flipping x and y. So locating 1 on the y-axis, so here is y equals 1. So we'll find then the corresponding x value so where our function hits this line y equals 1. Well, this happens at x equals 2. So that then is going to be the value of the inverse function at 1. So we found then that the inverse at 1 is 2.
Now, what this would also mean then is that if we located 2 on our x-axis, so this here is x equals 1, x equals 2, and go up to our function, we get 1. So to find the value of an inverse using a graph, we want to find the input value on the y-axis and find the corresponding x value.
So next, we're going to talk about finding the inverse algebraically. So we would like to write the inverse function to f of x equals 2x minus 3. Now, to start, I'm going to write this as y equals 2x minus 3 because we're going to return to the whole idea of swapping x and y.
Now, there are a couple of different ways you can go about this. One thing you could do is first swap x and y right away, and then rewrite the equation to express it as y equals. Another thing you can do is rewrite the equation to express as x equals, and then swap x and y last.
But the big idea here is it's going to involve swapping x and y. And then your final result is going to have y on one side of the equation, everything else on the other. So we're going to go about this using both methods.
So the first thing I'm going to do here with y equals 2x minus 3 is swap x and y. So I'm just going to write this as x equals 2y minus 3. Now, we need to isolate y. So we want to express this as y equals. So the first thing I need to do is add 3 to both sides of the equation. So x plus 3 equals 2y. And then I'll divide by 2. So y equals x plus 3 all over 2. So that is the inverse function.
Let's go ahead and do this using our other methods. So we're going to take y equals 2x minus 3. And we're going to express first as x equals. So I need to isolate x. This would involve adding 3 to both sides of my equation. So I have y plus 3 equals 2x. And then divide everything by 2. So y plus 3 all over 2 is x.
And the last thing I need to do is swap x and y. And you can see that this gives me the same equation. So writing this in function notation, I would say then that the inverse of x is x plus 3 over 2.
So we've established that when f of x equals 2x plus 3, it's inverse is x plus 3 all over 2. Let's go ahead then and improve that these are indeed inverses of each other by showing this relationship. So I'm going to take my inverse function. And the argument is going to be f of x. So instead of writing x, I'm going to write-- or instead of writing f of x, I'm going to write 2x minus 3.
So how I'm going to evaluate this now is I'm going to look at my expression for the inverse. And instead of writing x, I'm going to write 2x minus 3. So this gives me 2x minus 3 plus 3 all over 2. Well, I have a minus 3 and a plus 3 in the numerator. Those cancel. So my numerator simplifies to 2x. And 2x divided by 2 equals x. So you've proved then this relationship between a function and its inverse.
So let's review our lesson on finding the inverse of a function. Well, there's this relationship that if you use a function as the input for its inverse, you're going to get back x. So finding the inverse on a graph, what you could do is locate this a value on the y-axis of the function instead of the x-axis and find the corresponding x value.
So using this whole idea of swapping x and y, to find the inverse algebraically, you could take the function f of x, express it as y equals instead of f of x equals, swap x and y. And then just express your equation as y equals.
Another thing you could do is first isolate x on one side of the equation, but then swap x and y at the end. And that will give you the equation for the inverse. So thanks for watching this tutorial on finding the inverse of a function. Hope to see you next time.
