Chapter EP, Problem 7.1.3EP

To determine

To calculate: The value of $\left(f+g\right)\left(x\right),\left(f-g\right)\left(x\right),\left(f\cdot g\right)\left(x\right),\text{ and }\left(\frac{f}{g}\right)\left(x\right)$ where $f\left(x\right)=x^2-5\text{ and }g\left(x\right)=x^2+5$ .

Answer to Problem 7.1.3EP

The value of

  $\left(f+g\right)\left(x\right)=2x^2,\left(f-g\right)\left(x\right)=-10,\left(f\cdot g\right)\left(x\right)=x^4-25,\text{ and }\left(\frac{f}{g}\right)\left(x\right)=\frac{x^2-5}{x^2+5}$ .

Explanation of Solution

Given information:

The functions $f\left(x\right)=x^2-5\text{ and }g\left(x\right)=x^2+5$ .

Formula used:

The sum of two functions $f\left(x\right)\text{and }g\left(x\right)$ is $\left(f+g\right)\left(x\right)=f\left(x\right)+g\left(x\right)$ .

The difference of two functions $f\left(x\right)\text{and }g\left(x\right)$ is $\left(f-g\right)\left(x\right)=f\left(x\right)-g\left(x\right)$ .

The product of two functions $f\left(x\right)\text{and }g\left(x\right)$ is $\left(f\cdot g\right)\left(x\right)=f\left(x\right)\cdot g\left(x\right)$ .

The quotient of two functions $f\left(x\right)\text{and }g\left(x\right)$ is $\left(\frac{f}{g}\right)\left(x\right)=\frac{f\left(x\right)}{g\left(x\right)}$ .

Calculation:

Consider the functions $f\left(x\right)=x^2-5\text{ and }g\left(x\right)=x^2+5$ .

Recall that sum of two functions $f\left(x\right)\text{and }g\left(x\right)$ is $\left(f+g\right)\left(x\right)=f\left(x\right)+g\left(x\right)$ .

Apply it,

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

Recall that difference of two functions $f\left(x\right)\text{and }g\left(x\right)$ is $\left(f-g\right)\left(x\right)=f\left(x\right)-g\left(x\right)$ .

Apply it,

  $\begin{array}{c}\left(f-g\right)\left(x\right)=f\left(x\right)-g\left(x\right)\\ =x^2-5-\left(x^2+5\right)\\ =x^2-5-x^2-5\\ =-10\end{array}$

Recall that product of two functions $f\left(x\right)\text{and }g\left(x\right)$ is $\left(f\cdot g\right)\left(x\right)=f\left(x\right)\cdot g\left(x\right)$ .

Apply it,

  $\begin{array}{c}\left(f\cdot g\right)\left(x\right)=f\left(x\right)\cdot g\left(x\right)\\ =\left(x^2-5\right)\left(x^2+5\right)\\ =x^4+5x^2-5x^2-25\\ =x^4-25\end{array}$

Recall that quotient of two functions $f\left(x\right)\text{and }g\left(x\right)$ is $\left(\frac{f}{g}\right)\left(x\right)=\frac{f\left(x\right)}{g\left(x\right)}$ .

Apply it,

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

Thus, the value of

  $\left(f+g\right)\left(x\right)=2x^2,\left(f-g\right)\left(x\right)=-10,\left(f\cdot g\right)\left(x\right)=x^4-25,\text{ and }\left(\frac{f}{g}\right)\left(x\right)=\frac{x^2-5}{x^2+5}$ .