 In this video, we present the solution to question number 12 from practice exam number 1 for math 12-10. We're asked to find the inverse function f inverse of x, given f of x equals negative 2x cubed over x cubed minus 1. Inverse functions only exist for functions 1 to 1. You do not need to show that this is 1 to 1. You may just assume it. So what we're going to do is we're going to start off with our function f of x. So f of x equals negative 2x cubed over x cubed minus 1. Now the inverse function has the relationship that's going to reverse the rules of x and y. So it might be helpful instead to use the symbol y right now. So y equals negative 2x cubed over x cubed minus 1. And so this is a formula given for the function f right here. Now to switch to the inverse function f inverse there, we're going to swap the rules of x and y. So this y is going to become an x and these x's are going to become y. So we get negative 2y cubed over y cubed minus 1. Now we need to solve for y in this process. So to do that, since I have a y in the denominator here, I'm going to times both sides of the equation by y cubed minus 1. Thus clearing the denominator. So these are going to cancel out on the right hand side. So we're going to get x times y cubed minus 1 is equal to negative 2y cubed. At this moment, many students go down the wrong path of dividing both sides by negative 2 and then taking the cube root so that you get y equals on the right hand side. So yes, when you have to solve an equation for y, you do want to get y equals on one of the sides. But you have to first combine all the y's together. If there's a y on the other side, you didn't really solve for y yet. So what we need to do is we need to combine together the y's. Now to accomplish that, I need to free up the y which is in prison inside of these parentheses right here. You can do that by distribution. Distribute the x through which case we're going to get x times y cubed minus x. This is equal to negative 2y cubed. I now want to combine together the y's that are separated. So I'm going to add 2y cubed to both sides. And I'm going to add x so then only the multiples of y on the left hand side of this equation. So we get x times y cubed plus 2 times y cubed is equal to x. Now at this step, sometimes we get a little bit confused but you'll notice that everyone on the left hand side is a multiple of y cubed. As such, we can factor out the y cubed and this will then give us y cubed times x plus 2 is equal to x. You'll notice that by factoring actually got all of the y's together. Next, we need to get rid of the so-called coefficient of the y cube that is divide both sides by x plus 2 on both sides. So we end up with y cubed equals x over x plus 2. And then the last thing to do is to take the cube root of both sides. We end up with y is equal to the cube root of x plus x plus x over x plus 2. Now at this moment, we're not exactly done, right? We solve for the formula but it asks for f inverse. Where on the screen does it tell you what f inverse is? So you actually need to answer, make sure you answer the question, label it correctly. And so that could be an easy fix. If I just relabel the last part, f inverse of x, this is equal to the cube root of x over x plus 2. So we need to actually label who is f inverse. Don't just leave it for the grader to assume because the grader is to infer nothing, right? You need to tell the grader what the answer is. So make sure you do clear the label f inverse is this.