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