Find the inverse of given functions

Question

Suppose $ g(x)=\frac{x^3}{x^2+1} $ is the inverse of f(x). Find the inverse functions of the function f(x+1), and 4f(x).

I tried replacing all x’s with y and all y’s with x and had this: $ y^3-xy^2-x=0 $, but I don't know how to solve for y.

Answer 1

You don't need to find $f$; you just need the inverses of $f(x+1)$ and $4f(x)$. Therefore let's ignore the actual values of $g$ right now and look at this abstractly. If we think about it graphically, $f$ and $g$ are reflected across the line $y=x$; therefore if we move $f$ to the left by $1$ ($f(x+1)$) to find its inverse we move $g$ down by $1$: $g(x)-1$. Similarly, if $f$ is stretched vertically, we stretch $g$ horizontally: $g(x/4)$. Plugging the formula for $g$ into these we obtain our answers.