Answer
a) $f^{-1}(x)=x^3+1$
b) See graph
c) The graph of $f^{-1}$ is the reflection of the graph of $f$ across the line $y=x$.
d) $D_f=\left(-\infty,\infty\right),R_f=\left(-\infty,\infty\right)$
$D_{f^{-1}}=\left(-\infty,\infty\right),R_{f^{-1}}=\left(-\infty,\infty\right)$
Work Step by Step
We are given the function:
$f(x)=\sqrt[3]{x-1}$
$y=\sqrt[3]{x-1}$
a) Determine the inverse $f^{-1}$. Interchange $x$ and $y$:
$x=\sqrt[3]{y-1}$
$x^3=(\sqrt[3]{y-1})^3$
$x^3=y-1$
$y=x^3+1$
$f^{-1}(x)=x^3+1$
b) Graph both functions.
c) The graph of the function $f^{-1}$ is the reflection of the graph of $f$ across the line $y=x$.
d) Determine the domain and range of $f$:
$D_f=\left(-\infty,\infty\right)$
$R_f=\left(-\infty,\infty\right)$
Determine the domain and range of $f^{-1}$:
$D_{f^{-1}}=\left(-\infty,\infty\right)$
$R_{f^{-1}}=\left(-\infty,\infty\right)$
