Chapter SH, Problem 6.2EP

To determine

Find the solution using given arithmetic operation.

Answer to Problem 6.2EP

The results are given below.

  $\begin{array}{l}\left(f+g\right)\left(x\right)=x^2-x-1\\ \left(f-g\right)\left(x\right)=-x^2-x+7\\ \left(f·g\right)\left(x\right)=2x^2+4x-12\\ \left(\frac{f}{g}\right)\left(x\right)=\frac{-x+3}{x^2-4},x\ne \pm 2\end{array}$

Explanation of Solution

Given:

The given functionsare $f\left(x\right)=-x+3$ and $g\left(x\right)=x^2-4$ .

The given operations are $\left(f+g\right)\left(x\right)$ , $\left(f-g\right)\left(x\right)$ , $\left(f·g\right)\left(x\right)$ and $\left(\frac{f}{g}\right)\left(x\right)$ .

Calculation:

Apply addition of function.

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

Combine like terms.

  $\begin{array}{l}=x^2-x+\left(-4+3\right)\\ =x^2-x-1\end{array}$

Apply subtraction of function.

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

Combine like terms.

  $\begin{array}{l}=-x^2-x+\left(4+3\right)\\ =-x^2-x+7\end{array}$

Apply multiplication of function.

  $\begin{array}{l}\left(f·g\right)\left(x\right)=f\left(x\right)·g\left(x\right)\\ =\left(-x+3\right)\left(x^2-4\right)\end{array}$

Use distributive property.

  $\begin{array}{l}=-x\left(x^2-4\right)+3\left(x^2-4\right)\\ =-x^2+4x+3x^2-12\end{array}$

Combine like terms.

  $\begin{array}{l}=\left(-x^2+3x^2\right)+4x-12\\ =2x^2+4x-12\end{array}$

Apply division of function.

  $\begin{array}{l}\left(\frac{f}{g}\right)\left(x\right)=\frac{f\left(x\right)}{g\left(x\right)},g\left(x\right)\ne 0\\ =\frac{-x+3}{x^2-4},x\ne \pm 2\end{array}$

Hence the results are given below.

  $\begin{array}{l}\left(f+g\right)\left(x\right)=x^2-x-1\\ \left(f-g\right)\left(x\right)=-x^2-x+7\\ \left(f·g\right)\left(x\right)=2x^2+4x-12\\ \left(\frac{f}{g}\right)\left(x\right)=\frac{-x+3}{x^2-4},x\ne \pm 2\end{array}$