College Algebra, Chapter 3, 3.7, Section 3.7, Problem 2

a.) For the function to have an inverse, it must be $\underline{\text{a one to one function}}$. So, which one of the following functions has an inverse?

$f(x) = x^2 \qquad g(x) = x^3$

b.) What is the inverse of the function you chose in part (a).

a.) Since, the function $f(x) = x^2$ is increasing and is symmetric to $y$-axis, then if you use horizontal line test, the function will intersect the line more than once, that's why $f(x) = x^2$ is not one to one. On the other hand, $g(x) = x^3$ is increasing and is symmetric to origin, that's why it has an inverse. Thus, $g(x) = x^3$ is one to one.

b.) To find the inverse, first, we write $y = g(x)$

$y = x^3$

Then solve for $x$,


$\begin{equation} \begin{aligned} x =& \sqrt[3]{y} \qquad \text{Interchange $y$ and $x$} \\ \\ y =& \sqrt[3]{x} \end{aligned} \end{equation}$


Thus, the inverse of $g(x) = x^3$ is $h(x) = x^{\frac{1}{3}}$