Answer
(a) The inverse function is $~~g^{-1}(x) = \sqrt[3] {1-x^3}$
$g^{-1}(x) = g(x)$
(b) If we reflect the graph of $g(x)$ in the line $y=x$, we will get exactly the same graph. The function $g(x)$ is its own inverse.
Work Step by Step
(a) We can solve $g$ for $x$:
$y = \sqrt[3] {1-x^3}$
$y^3 = 1-x^3$
$y^3-1 = -x^3$
$x^3 = 1-y^3$
$x = \sqrt[3] {1-y^3}$
We reverse the places of $x$ and $y$:
$y = \sqrt[3] {1-x^3}$
The inverse function is $~~g^{-1}(x) = \sqrt[3] {1-x^3}$
$g^{-1}(x) = g(x)$
(b) If we reflect the graph of $g(x)$ in the line $y=x$, we will get exactly the same graph. The function $g(x)$ is its own inverse.
