Answer
The pair of provided functions are not inverses of each other.
Work Step by Step
$f\left( x \right)=0.8{{x}^{1/2}}+5.23$and$g\left( x \right)=1.25\left( {{x}^{2}}-5.23 \right),\text{ }x\ge 0$
Now apply the formula of composition of two functions
$f\circ g\left( x \right)=f\left( g\left( x \right) \right)$
Substitute $1.25\left( {{x}^{2}}-5.23 \right)$ for $g\left( x \right)$ in the above equation
$f\circ g\left( x \right)=f\left( 1.25\left( {{x}^{2}}-5.23 \right) \right)$
Substitute $1.25\left( {{x}^{2}}-5.23 \right)$ for $x$ in the provided function $f\left( x \right)$; the above equation becomes
$f\circ g\left( x \right)=0.8{{\left( 1.25\left( {{x}^{2}}-5.23 \right) \right)}^{1/2}}+5.23$
Use the distributive property in the above equation
$\begin{align}
& f\circ g\left( x \right)=0.8{{\left( 1.25\times {{x}^{2}}-1.25\times 5.23 \right)}^{1/2}}+5.23 \\
& =0.8{{\left( 1.25{{x}^{2}}-6.5375 \right)}^{1/2}}+5.23
\end{align}$
Here,
$f\circ g\left( x \right)\ne x$
Here, the composition of two functions is not the identity function.
