Answer
(a) The function $g(x)=\sqrt[3] {1-x^{3}}$ and
$g^{-1}(x)=\sqrt[3] {1-x^{3}}$is same.
(b) Both the functions $f(x)$ and $f^{-1}(x)$ represent the same graph in the positive quadrant as depicted below:
Work Step by Step
(a) Calculate the inverse of the function $g(x)=\sqrt[3] {1-x^{3}}$
Write $y=g(x)$
$y=\sqrt[3] {1-x^{3}}$
Solve this equation for x in terms of y to get the inverse function.
$y^{3}=1-x^{3}$
$x^{3} + y^{3} =1$
$x=\sqrt[3] {1-y^{3}}$
To express $g^{-1}(x)$ as a function of x,interchange x and y. The resulting equation is
$y=\sqrt[3] {1-x^{3}}$
Therefore, the inverse of the function $g^{-1}(x)=y=\sqrt[3] {1-x^{3}}$
Hence, the function $g(x)=\sqrt[3] {1-x^{3}}$ and
$g^{-1}(x)=\sqrt[3] {1-x^{3}}$is same.
(b) Solve this equation for x in terms of y to get the inverse function.
$y^{3}=1-x^{3}$
$x^{3} + y^{3} =1$
$x=\sqrt[3] {1-y^{3}}$
Both the functions $f(x)$ and $f^{-1}(x)$ represent the same graph in the positive quadrant as depicted below:
