College Algebra, Chapter 3, Review Exercises, Section Review Exercises, Problem 92

a.) Sketch the graph of $f$
b.) Show that the function $f(x) = 1 + \sqrt[4]{x}$ is one to one.
c.) Use part(a) to sketch the graph of $f^{-1}$
d.) Find an equation of $f^{-1}$

a.)



b.) It shows from the graph of $f$ that the function is a one to one function since there is only one corresponding $y$-value for each value of $x$. Also, if you use Horizontal Line Test, the line will intersect the function only once.

c.) To sketch the graph of $f^{-1}$ we reflect the graph of $f$ about the line $y = x$



d.) To find the equation of $f^{-1}$, we set $y = f(x)$

$\begin{equation} \begin{aligned} y &= 1 + \sqrt[4]{x} && \text{Solve for } x \text{, subtract } 1\\ \\ \sqrt[4]{x} &= y - 1 && \text{Raise both sides by 4}\\ \\ x &= (y - 1)^4 && \text{Interchange } x \text{ and } y\\ \\ y &= (x-1)^4 \end{aligned} \end{equation}$

Thus, $f^{-1}(x) = (x-1)^4$ for $x > 1$