Show algebraically that the functions

f

and

g

shown below are inverse functions.

f ( x ) = \sqrt [ 3 ] { 8 x - 7 } , \ g ( x ) = \frac { x ^ { 3 } + 7 } { 8 }

A)

\begin{aligned}f(g(x)) &=\sqrt[3]{8\left(\frac{x^{3}+7}{8}\right)-7} \\&=\sqrt[3]{\left(x^{3}+56\right)-56} \\&=\sqrt[3]{x^{3}+56-56} \\&=\sqrt[3]{x^{3}} \\&=x\end{aligned}

\begin{aligned}g(f(x)) &=\frac{(\sqrt[3]{8 x-7})^{3}+7}{8} \\&=\frac{8 x-7^{3}+7^{3}}{8} \\&=\quad \frac{8 x}{8}\\&=x\end{aligned}

B)

\begin{aligned}f(g(x)) &=\sqrt[3]{8\left(\frac{x^{3}+7}{8}\right)-7} \\&=\sqrt[3]{\left(x^{3}+7\right)-7} \\&=\sqrt[3]{x^{3}+7-7} \\&=\sqrt[3]{x^{3}} \\&=x\end{aligned}

\begin{aligned}g(f(x)) &=& \frac{(\sqrt[3]{8 x-7})^{3}+7}{8} \\&=& \frac{8 x-7+7}{8} \\&=& \frac{8 x}{8} \\&=& x\end{aligned}

C)

\begin{aligned}f(g(x)) &=\sqrt[3]{8\left(\frac{x^{3}+7}{8}\right)-7} \\&=\sqrt[3]{\left(\frac{8 x^{3}+7}{8}\right)-7} \\&=\sqrt[3]{x^{3}+7-7} \\&=\sqrt[3]{x^{3}} \\&=x\end{aligned}

\begin{aligned}g(f(x)) &=\frac{(\sqrt[3]{8 x-7})^{3}+7}{8} \\&=\frac{8^{3} x-7+7}{8^{3}} \\&=\frac{8^{3} x}{8^{3}} \\&=x\end{aligned}

D)

\begin{aligned}f(g(x)) &=\sqrt[3]{8\left(\frac{x^{3}+7}{8}\right)-7} \\&=\sqrt[3]{\left(8 x^{3}+56\right)-56} \\&=\sqrt[3]{8 x^{3}+56-56} \\&=\sqrt[3]{8 x^{3}} \\&=x\end{aligned}

\begin{aligned}g(f(x)) &=\frac{(\sqrt[3]{8 x-7})^{3}+7}{8} \\&=\frac{8^{3} x-7^{3}+7^{3}}{8^{3}} \\&=\frac{8^{3} x}{8^{3}} \\&=\quad x\end{aligned}

E)

\begin{aligned}f(g(x)) &=\sqrt[3]{8\left(\frac{x^{3}+7}{8}\right)-7} \\&=\sqrt[3]{\left(x^{3}+\frac{7}{8}\right)-7} \\&=\sqrt[3]{x^{3}+\frac{0}{8}} \\&=\sqrt[3]{x^{3}} \\&=x\end{aligned}

\begin{aligned}g(f(x)) &=\frac{(\sqrt[3]{8 x-7})^{3}+7}{8} \\&=\frac{24 x-21+21}{24} \\&=\quad \frac{24 x}{24} \\&=\quad x\end{aligned}

Given f(x)=10x2−9f ( x ) = \frac { 10 } { x ^ { 2 } - 9 }f(x)=x2−910​

Multiple Choice