 In this video, we present the solution to question number 12 for practice exam number three for math 1050 We're given the one-to-one function f of x equals negative 2x cubed over x cubed minus one and we have to find the inverse function So remember the business with the inverse function I'm going to remove the notation f of x and actually think of the variables x and y right here So you have y equals negative 2x cubed Over x cubed minus one this of course gives us the graph of f We then want to switch to the graph of f inverse And we do that by switching the x and y variables So the y becomes an x and all of the x's become y so you get negative 2 y cubed over Y cubed minus one it's now our goal to solve For the y variable since we have a y in the numerator and the denominator We're going to have to clear the denominators to try to combine the y's together So i'm going to times the left hand side by y cubed minus one so that they cancel But what's good for the goose is good for the gander. We times the left hand side by y cubed minus one as well This gives us x times y cubed minus one the right hand side will just be negative 2 y cubed of course We're then going to free up the y because again, we want to get the y cubes together The y is now trapped inside the parentheses So we have to distribute the x so that we can drop the parentheses from consideration This gives us x y cubed minus x is equal to negative 2 y cubed So now we're in a situation where we can combine the y cubes together So i'm going to move the negative 2 y cube to the left hand side by adding 2 y cubed to both sides I'm going to move the negative x to the right hand side by adding x to both sides This then gives us x y cubed plus 2 y cubed is equal to positive x Now on the left hand side, you'll notice that everyone is a multiple of y cubed. We can factor it out Thus giving us y cubed times x plus 2 this is equal to x Then to solve for the y cubed we'll divide both sides by x plus 2 x plus 2 This gives us y cubed is equal to x over x plus 2 So we've successfully solved for the y cubed But there's some there's some numbers still attached to the y in particular in this example We still have the y cubed we got to get rid of the third power The inverse operation there would be the cube root So take the cube root of both sides that'll then give you that y equals the cube root of x Over x plus 2 and therefore we can then record our inverse function F inverse of x is the cube root of x over x plus 2 Make sure that that cube root reaches all the way to the bottom of the fraction Both the numerator x and the denominator x plus 2 are within the scope of that cube root And this then gives us this inverse function cube root of x over x plus 2