Answer
a) $f^{-1}(x)=\sqrt{4-x^2},x\in [0,2]$
b) See graph
c) The graph of $f^{-1}$ is the reflection of the graph of $f$ across the line $y=x$
d) $D_f=[0,2],R_f=[0,2]$
$D_{f^{-1}}=[0,2],R_{f^{-1}}=[0,2]$
Work Step by Step
We are given the function:
$f(x)=\sqrt{4-x^2},x\in [0,2]$
$y=\sqrt{4-x^2}$
a) Determine the inverse $f^{-1}$. Interchange $x$ and $y$:
$x=\sqrt{4-y^2}$
$x^2=(\sqrt{4-y^2})^2$
$x^2=4-y^2$
$y^2=4-x^2$
$y=\sqrt{4-x^2}$
$f^{-1}(x)=\sqrt{4-x^2},x\in [0,2]$
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 $f$:
$D_f=[0,2]$
$R_f=[0,2]$
Determine the domain and range of $f^{-1}$:
$D_{f^{-1}}=[0,2]$
$R_{f^{-1}}=[0,2]$
