Question

How do you find the inverse of `f(x)=x^3-2` and is it a function?

Answer 1

`f^-1(x)=color(white)(0)^3sqrtx+2`

Explanation:

We begin with `f(x)=x^3-2`. To find the inverse of any equation, just switch `x` and `y`. No, before we do thta, I'm going to change the equation. I'm going to rename `f(x)` to `y`, just so that I don't have to deal with the parentheses or anything. Now, all I'bve doe is change what `f(x)` is called, not it's actual value.

Anyways, we have `y=x^3-2`. Now we switch `x` and `y` and then solve for `y`.

Now we've got `x=y^3-2`. If we add `2` on both sides we have `x+2=y^3`. Now we just need to get `y` as simple as possible, which we'll do by cube rooting both sides of the equation. That leaves us with `y=color(white)(0)^3sqrtx+2`. Now we just change `y` back to `f(x)` and add a `color(white)(0)^-1` to write it in inverse notation, and we have `f^-1(x)=color(white)(0)^3sqrtx+2`.