Chapter 1, Problem 25SGA

To determine

To find: $\left[f\circ g\right]\left[x\right]$and $\left[g\circ f\right]\left[x\right]$ .

Answer to Problem 25SGA

  $\begin{array}{l}\left[f\circ g\right]\left[x\right]=1.5x^2+5\\ \left[g\circ f\right]\left[x\right]=0.75x^2+15x+75\end{array}$

Explanation of Solution

Given information: The functions given are as follows:

  $\begin{array}{l}f\left(x\right)=0.5x+5\\ g\left(x\right)=3x^2\end{array}$

Calculation:

Put the values of $g\left(x\right)$ in $f\left(g\left(x\right)\right)$ in the equation below and simplify as follows:

  $\begin{array}{c}\left[f\circ g\right]\left[x\right]=f\left(g\left(x\right)\right)\\ =0.5\left(g\left(x\right)\right)+5\\ =\left(0.5\right)\left(3x^2\right)+5\left[\text{Substitute }g\left(x\right)\right]\\ =1.5x^2+5\end{array}$

Thus, the value of $\left[f\circ g\right]\left[x\right]$ is $1.5x^2+5$ .

Put the values of $f\left(x\right)$ in $g\left(f\left(x\right)\right)$ in the equation below and simplify as follows:

  $\begin{array}{c}\left[g\circ f\right]\left[x\right]=g\left(f\left(x\right)\right)\\ =3{\left(f\left(x\right)\right)}^2\\ =3{\left(0.5x+5\right)}^2\left[\text{Substitute }f\left(x\right)\right]\\ =0.75x^2+15x+75\end{array}$

Thus, the value of $\left[g\circ f\right]\left[x\right]$ is $0.75x^2+15x+75$ .