Answer
The pair of provided functions are inverses of each other.
Work Step by Step
$f\left( x \right)=2.5\left( {{x}^{3}}-7.1 \right)$and$g\left( x \right)=\sqrt[3]{0.4x+7.1}$
Now apply the formula for the composition of two functions
$f\circ g\left( x \right)=f\left( g\left( x \right) \right)$
Substitute$\sqrt[3]{0.4x+7.1}$ for$g\left( x \right)$ in the above equation
$f\circ g\left( x \right)=f\left( \sqrt[3]{0.4x+7.1} \right)$
Substituting $\sqrt[3]{0.4x+7.1}$ for $x$ in the provided function $f\left( x \right)$, the above equation becomes
$\begin{align}
& f\circ g\left( x \right)=2.5\left( {{\left( \sqrt[3]{0.4x+7.1} \right)}^{3}}-7.1 \right) \\
& =2.5\left( 0.4x+7.1-7.1 \right) \\
& =2.5\left( 0.4x \right) \\
& =x
\end{align}$
Since the compositions equal x, the functions are inverses.
