Chapter 6.1, Problem 9PPS

To determine

The value of $\left(f+g\right)\left(x\right),\left(f-g\right)\left(x\right)\left(f•g\right)\left(x\right)\text{and }\left(\frac{f}{g}\right)\left(x\right)$ for the pair of functions $f\left(x\right)=x-1\text{ and }g\left(x\right)=5x-2$ .

Answer to Problem 9PPS

The value $\left(f+g\right)\left(x\right)=6x-3,\left(f-g\right)\left(x\right)=-4x+1,\left(f•g\right)\left(x\right)=5x^2-7x+2$ and $\left(\frac{f}{g}\right)\left(x\right)=\frac{x-1}{5x-2};x\ne \frac{2}{5}$ for the pair of functions $f\left(x\right)=x-1\text{ and }g\left(x\right)=5x-2$ .

Explanation of Solution

Given information: The given functions are,

  $\left(\begin{array}{l}f\left(x\right)=x-1\\ g\left(x\right)=5x-2\end{array}\right)\mathrm{......}\left(1\right)$

Calculation: Evaluating $\left(f+g\right)\left(x\right)$ using (1),

  $\begin{array}{c}\left(f+g\right)\left(x\right)=\left(x-1\right)+\left(5x-2\right)\\ =6x-3\end{array}$

Evaluating $\left(f-g\right)\left(x\right)$ using (1),

  $\begin{array}{c}\left(f-g\right)\left(x\right)=f\left(x\right)-g\left(x\right)\\ =\left(x-1\right)-\left(5x-2\right)\\ =-4x+1\end{array}$

Now, evaluating $\left(f•g\right)\left(x\right)$ using (1).

  $\begin{array}{c}\left(f•g\right)\left(x\right)=f\left(x\right)\times g\left(x\right)\\ =\left(x-1\right)\left(5x-2\right)\\ =5x^2-7x+2\end{array}$

Similarly evaluating $\left(\frac{f}{g}\right)\left(x\right)$ using (1).

  $\begin{array}{c}\left(\frac{f}{g}\right)\left(x\right)=\frac{f\left(x\right)}{g\left(x\right)}\mathrm{......}\left(\text{where }g\left(x\right)\ne 0\right)\\ =\frac{x-1}{5x-2}\mathrm{.....}\left(\text{For }x\ne \frac{2}{5}\right)\end{array}$