Chapter 1.5, Problem 14E

(a)

To determine

To calculate: To find $\left(f+g\right)\left(x\right)$

Answer to Problem 14E

  $\left(f+g\right)\left(x\right)=x^2+2x-4$

Explanation of Solution

Given: $f\left(x\right)=2x+5$ and $g\left(x\right)=x^2-9$

Formula Used: Addition of two functions $f$ and $g$ is defined as

  $\left(f+g\right)\left(x\right)=f\left(x\right)+g\left(x\right)$

Calculation:

Functions are given as

  $f\left(x\right)=2x+5$ and $g\left(x\right)=x^2-9$

  $\left(f+g\right)\left(x\right)$ is calculated as follows:

  $\left(f+g\right)\left(x\right)=f\left(x\right)+g\left(x\right)$

Substituting the values:

  $\begin{array}{l}\left(f+g\right)\left(x\right)=f\left(x\right)+g\left(x\right)\\ \left(f+g\right)\left(x\right)=\left(2x+5\right)+\left(x^2-9\right)\\ \left(f+g\right)\left(x\right)=x^2+2x-4\end{array}$

Conclusion:

Hence, $\left(f+g\right)\left(x\right)=x^2+2x-4$

(b)

To determine

To calculate: To find $\left(f-g\right)\left(x\right)$

Answer to Problem 14E

  $\left(f-g\right)\left(x\right)=-x^2+2x+14$

Explanation of Solution

Given: $f\left(x\right)=2x+5$ and $g\left(x\right)=x^2-9$

Formula Used: Subtraction of two functions $f$ and $g$ is defined as

  $\left(f-g\right)\left(x\right)=f\left(x\right)-g\left(x\right)$

Calculation:

Functions are given as

  $f\left(x\right)=2x+5$ and $g\left(x\right)=x^2-9$

  $\left(f-g\right)\left(x\right)$ is calculated as follows:

  $\left(f-g\right)\left(x\right)=f\left(x\right)-g\left(x\right)$

Substituting the values:

  $\begin{array}{l}\left(f-g\right)\left(x\right)=f\left(x\right)-g\left(x\right)\\ \left(f-g\right)\left(x\right)=\left(2x+5\right)-\left(x^2-9\right)\\ \left(f-g\right)\left(x\right)=-x^2+2x+14\end{array}$

Conclusion:

Hence, $\left(f-g\right)\left(x\right)=-x^2+2x+14$

(c)

To determine

To calculate: To find $\left(fg\right)\left(x\right)$

Answer to Problem 14E

  $\left(fg\right)\left(x\right)=2x^3+5x^2-18x-45$

Explanation of Solution

Given: $f\left(x\right)=2x+5$ and $g\left(x\right)=x^2-9$

Formula Used: Product of two functions $f$ and $g$ is defined as

  $\left(fg\right)\left(x\right)=f\left(x\right)g\left(x\right)$

Calculation:

Functions are given as

  $f\left(x\right)=2x+5$ and $g\left(x\right)=x^2-9$

  $\left(fg\right)\left(x\right)$ is calculated as follows:

  $\left(fg\right)\left(x\right)=f\left(x\right)g\left(x\right)$

Substituting the values:

  $\begin{array}{l}\left(fg\right)\left(x\right)=\left(2x+5\right)\left(x^2-9\right)\\ \left(fg\right)\left(x\right)=2x^3-18x+5x^2-45\\ \left(fg\right)\left(x\right)=2x^3+5x^2-18x-45\end{array}$

Conclusion:

Hence, $\left(fg\right)\left(x\right)=2x^3+5x^2-18x-45$

(d)

To determine

To calculate: To find $\left(f/g\right)\left(x\right)$

Answer to Problem 14E

  $\left(f/g\right)\left(x\right)=\frac{2x+5}{x^2-9}$

Explanation of Solution

Given: $f\left(x\right)=2x+5$ and $g\left(x\right)=x^2-9$

Formula Used: Quotient of two functions $f$ and $g$ is defined as

  $\left(f/g\right)\left(x\right)=\frac{f\left(x\right)}{g\left(x\right)}$

Calculation:

Functions are given as

  $f\left(x\right)=2x+5$ and $g\left(x\right)=x^2-9$

  $\left(f/g\right)\left(x\right)$ is calculated as follows:

  $\left(f/g\right)\left(x\right)=\frac{f\left(x\right)}{g\left(x\right)}$

Substituting the values:

  $\left(f/g\right)\left(x\right)=\frac{2x+5}{x^2-9}$

To calculate domain of $\left(f/g\right)\left(x\right)$ :

Denominator can never be zero

  $\begin{array}{l}{x}^2-9\ne 0\\ x^2\ne 9\\ x\ne -3,3\end{array}$

Thus, domain is $\left(-\infty ,\infty \right)-\left\{-3,3\right\}$

Conclusion:

Hence, $\left(f/g\right)\left(x\right)=\frac{2x+5}{x^2-9}$ and domain is all set of real numbers except $-3,3$