Finding inverses of rational functions | Equations | Algebra 2 | Khan Academy

All right, let's say that we have the function f of x and it's equal to 2x plus 5 over 4 minus 3x. What we want to do is figure out what is the inverse of our function. Pause this video and try to figure that out before we work on that together.

All right, now let's work on it together. Just as a reminder of what a function and an inverse even does: if this is the domain of a function and that's the set of all values that you could input into the function for x and get a valid output.

Let's say you have an x here. It's a member of the domain, and if I were to apply the function to it, if I were to input that x into that function, then the function is going to output a value in the range of the function, and we call that value f of x.

Now, an inverse goes the other way. If you were to input the f of x value into the function, that's going to take us back to x. So, that's exactly what f inverse does.

Now, how do we actually figure out the inverse of a function, especially a function that's defined with a rational expression like this? Well, the way that I think about it is, let's say that y is equal to our function of x, or y is a function of x. So we could say that y is equal to (2x + 5) / (4 - 3x).

For our inverse, the relationship between x and y is going to be swapped. In our inverse, it's going to be true that x is equal to (2y + 5) / (4 - 3y).

Then, to be able to express this as a function of x, to say that what is y as a function of x for our inverse, we now have to solve for y. So it's just a little bit of algebra here.

So let's see if we can do that. The first thing that I would do is multiply both sides of this equation by (4 - 3y). If we do that on the left-hand side, we are going to get x times each of these terms. So we're going to get 4x - 3yx, and then that's going to be equal to, on the right-hand side, since we multiplied by the denominator here, we're just going to be left with the numerator. It's going to be equal to 2y + 5.

This could be a little bit intimidating because we're seeing x's and y's. What are we trying to do? Remember, we're trying to solve for y.

So let's gather all the y terms on one side and all the non-y terms on the other side. Let's get rid of this 2y here, actually. Well, I could go either way. Let's get rid of this 2y here, so let's subtract 2y from both sides.

Then let's get rid of this 4x from the left-hand side, so let's subtract 4x from both sides. What are we going to be left with on the left-hand side? We're left with minus or negative 5. Oh, actually it would be this way: it would be negative 3yx - 2y.

You might say, "Hey, where is this going?" but I'll show you in a second. It is equal to those cancel out, and we're going to have 5 - 4x.

Now, once again, we are trying to solve for y. So let's factor out a y here. Then we are going to have y times (-3x - 2) = 5 - 4x.

Now this is the home stretch. We can just divide both sides of this equation by (-3x - 2), and we're going to get y = (5 - 4x) / (-3x - 2).

Now, another way that you could express this: you could multiply both the numerator and the denominator by -1. That won't change the value, and then you would get, in the numerator, 4x - 5, and in the denominator, you would get 3x + 2.

So there you have it: our f inverse as a function of x, which you could say is equal to this y = (4x - 5) / (3x + 2).