Answer
a) $f^{-1}(x)=\sqrt[3]{x-8}$
b) See graph
c) The graph of $f^{-1}$ is the reflection of the graph of $f$ across the line $y=x$
d) $D_f=(-\infty,\infty),R_f=(-\infty,\infty)$
$D_{f^{-1}}=(-\infty,\infty),R_{f^{-1}}=(-\infty,\infty)$
Work Step by Step
We are given the function:
$f(x)=x^3+8$
$y=x^3+8$
a) Determine the inverse $f^{-1}$. Interchange $x$ and $y$:
$x=y^3+8$
$y^3=x-8$
$y=\sqrt[3]{x-8}$
$f^{-1}(x)=\sqrt[3]{x-8}$
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 $f$:
$D_f=(-\infty,\infty)$
$R_f=(-\infty,\infty)$
Determine the domain and range of $f^{-1}$:
$D_{f^{-1}}=(-\infty,\infty)$
$R_{f^{-1}}=(-\infty,\infty)$
