Question

How do you find the inverse function of `f(x)=x^3+5`?

Answer 1

`f^{-1}(x)=root(3)(x-5)`

Explanation:

Set `y=f(x)=x^3+5` and solve for `x`: `x^3=y-5\rightarrow x=root(3)(y-5)`.

It's okay to write the answer as `x=f^{-1}(y)=root(3)(y-5)`. It's also okay to swap variables and write the answer as `y=f^{-1}(x)=root(3)(x-5)`.

Swapping the variables is a good idea when you want to graph both functions in the same picture while seeing the "reflection property" of a function and its inverse across the line `y=x`. Not swapping the variables is a good idea if they have a real life meaning (which you would "lose" by swapping them).

Do note that the original function is indeed a one-to-one function since its graph passes the horizontal line test (see the graph below). There's no need to restrict its domain in any way.

graph{x^3+5 [-40, 40, -20, 20]}