Answer
The value of the function $\left( f\circ g \right)\left( x \right)$ is $4{{x}^{2}}-12x+10$.
The value of the function $\left( g\circ f \right)\left( x \right)$ is $2{{x}^{2}}-1$.
Work Step by Step
$f\left( x \right)={{x}^{2}}+1$ and $g\left( x \right)=2x-3$
Evaluate the value of the function $\left( f\circ g \right)\left( x \right)$ as follows.
$\left( f\circ g \right)\left( x \right)=f\left( g\left( x \right) \right)$
Use the function $g\left( x \right)=2x-3$ as follows.
$\left( f\circ g \right)\left( x \right)=f\left( 2x-3 \right)$
Use the function $f\left( x \right)={{x}^{2}}+1$ as follows.
$\begin{align}
& \left( f\circ g \right)\left( x \right)={{\left( 2x-3 \right)}^{2}}+1 \\
& =4{{x}^{2}}-12x+9+1 \\
& =4{{x}^{2}}-12x+10
\end{align}$
Thus, the value of the function $\left( f\circ g \right)\left( x \right)$ is $4{{x}^{2}}-12x+10$.
Evaluate the value of the function $\left( g\circ f \right)\left( x \right)$ as follows.
$\left( g\circ f \right)\left( x \right)=g\left( f\left( x \right) \right)$
Use the function $f\left( x \right)={{x}^{2}}+1$ as follows.
$\left( g\circ f \right)\left( x \right)=g\left( {{x}^{2}}+1 \right)$
Use the function $g\left( x \right)=2x-3$ as follows.
$\begin{align}
& \left( g\circ f \right)\left( x \right)=2\left( {{x}^{2}}+1 \right)-3 \\
& =2{{x}^{2}}+2-3 \\
& =2{{x}^{2}}-1
\end{align}$
Thus, the value of $\left( g\circ f \right)\left( x \right)$ is $2{{x}^{2}}-1$.
