Answer
a) $f^{-1}(x)=x^{5/3}$
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:
$f(x)=x^{3/5}$
$y=x^{3/5}$
a) Determine the inverse $f^{-1}$. Interchange $x$ and $y$:
$x=y^{3/5}$
$x^{5/3}=(y^{3/5})^{5/3}$
$x^{5/3}=y$
$y=x^{5/3}$
$f^{-1}(x)=x^{5/3}$
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 $f^{-1}$:
$D_{f^{-1}}=\left(-\infty,\infty\right)$
$R_{f^{-1}}=\left(-\infty,\infty\right)$
