Chapter SH, Problem 6.3EP

To determine

Find the solution using given operation.

Answer to Problem 6.3EP

The results are given below.

  $\begin{array}{l}\left(f\circ g\right)\left(x\right)=3x-4\\ \left(g\circ f\right)\left(x\right)=3x-1\end{array}$

Explanation of Solution

Given:

The given functions are $f\left(x\right)=3x+2$ and $g\left(x\right)=x-3$ .

The given operations are $\left(f\circ g\right)\left(x\right)$ and $\left(g\circ f\right)\left(x\right)$ .

Calculation:

Apply property.

  $\begin{array}{l}\left(f\circ g\right)\left(x\right)=f\left[g\left(x\right)\right]\\ f\left(x\right)=3x+2\end{array}$

Substitute the value of $x=x-3$ .

  $\begin{array}{l}\left(f\circ g\right)\left(x\right)\\ =f\left[x-3\right]\\ =3\left(x-3\right)+2\\ =3x-6+2\\ =3x-4\end{array}$

Apply property.

  $\begin{array}{l}\left(g\circ f\right)\left(x\right)=g\left[f\left(x\right)\right]\\ g\left(x\right)=x-3\end{array}$

Substitute the value of $x=3x+2$ .

  $\begin{array}{l}\left(f\circ g\right)\left(x\right)\\ =g\left[3x+2\right]\\ =3x+2-3\\ =3x-1\end{array}$

Hence the results are given below.

  $\begin{array}{l}\left(f\circ g\right)\left(x\right)=3x-4\\ \left(g\circ f\right)\left(x\right)=3x-1\end{array}$