Calculus 8th Edition

(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:
(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: