Chapter D, Problem 1DE

Given $f\left(x\right)=x^2+3$ and $g\left(x\right)=x^2-3$ find each of the following.

a. $\left(f+g\right)\left(x\right)$

b. $\left(f-g\right)\left(x\right)$

c. $\left(f\cdot g\right)\left(x\right)$

d. $\left(f/g\right)\left(x\right)$

e. $\left(f\cdot f\right)\left(x\right)$

To determine

To calculate: The expression which represents $\left(f+g\right)x$ for the functions, $f\left(x\right)=x^2+3,g\left(x\right)=x^2-3$.

Answer to Problem 1DE

Solution:

The required expression for the provided functions $f\left(x\right)=x^2+3,g\left(x\right)=x^2-3$ is $\left(f+g\right)x=2x^2$.

Explanation of Solution

Given information:

Two functions,

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

Formula used:

Addition of two functions,

If there are two functions $f\left(x\right),g\left(x\right)$.

Then,

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

Calculation:

Consider the functions provided.

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

Use the principal of addition of two functions if the functions are $f\left(x\right),g\left(x\right)$.

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

Therefore,

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

Hence, the addition of the functions is $\left(f+g\right)x=2x^2$.

To determine

To calculate: The expression which represents $\left(f-g\right)x$ for the functions, $f\left(x\right)=x^2+3,g\left(x\right)=x^2-3$.

Answer to Problem 1DE

Solution:

The required expression for the provided functions $f\left(x\right)=x^2+3,g\left(x\right)=x^2-3$ is $\left(f-g\right)x=6$.

Explanation of Solution

Given information:

Two functions,

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

Formula used:

Subtraction of two functions,

If there are two functions $f\left(x\right),g\left(x\right)$.

Then,

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

Calculation:

Consider the functions provided.

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

Use the principal of subtraction of two functions if the functions are $f\left(x\right),g\left(x\right)$.

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

Therefore,

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

Hence, the required expression for the provided functions $f\left(x\right)=x^2+3,g\left(x\right)=x^2-3$ is $\left(f-g\right)x=6$.

To determine

To calculate: The expression which represents $\left(f\cdot g\right)x$ for the functions, $f\left(x\right)=x^2+3,g\left(x\right)=x^2-3$.

Answer to Problem 1DE

Solution:

The required expression for the provided functions $f\left(x\right)=x^2+3,g\left(x\right)=x^2-3$ is $\left(f\cdot g\right)x=x^4-9$.

Explanation of Solution

Given information:

Two functions,

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

Formula used:

Multiplication of two functions.,

If there are two functions $f\left(x\right),g\left(x\right)$.

Then,

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

Calculation:

Consider the functions provided.

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

Use the principal of multiplication of two functions if the functions are $f\left(x\right),g\left(x\right)$.

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

Therefore,

$\begin{array}{c}\left(f\cdot g\right)x=f\left(x\right)\cdot g\left(x\right)\\ =\left(x^2+3\right)\left(x^2-3\right)\\ \left(f\cdot g\right)x=x^4-9\end{array}$

Hence, the required expression for the provided functions $f\left(x\right)=x^2+3,g\left(x\right)=x^2-3$ is $\left(f\cdot g\right)x=x^4-9$.

To determine

To calculate: The expression which represents $\left(\raisebox{1ex}{$f$}\!\left/ \!\raisebox{-1ex}{$g$}\right.\right)x$ for the functions, $f\left(x\right)=x^2+3,g\left(x\right)=x^2-3$.

Answer to Problem 1DE

Solution:

The required expression for the provided functions $f\left(x\right)=x^2+3,g\left(x\right)=x^2-3$ is $\left(\raisebox{1ex}{$f$}\!\left/ \!\raisebox{-1ex}{$g$}\right.\right)x=\frac{x^2+3}{x^2-3}$.

Explanation of Solution

Given information:

Two functions,

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

Formula used:

Division of two functions.,

If there are two functions $f\left(x\right),g\left(x\right)$.

Then,

$\left(\raisebox{1ex}{$f$}\!\left/ \!\raisebox{-1ex}{$g$}\right.\right)x=\frac{f\left(x\right)}{g\left(x\right)}$

Calculation:

Consider the functions provided,

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

Use the principal of division of two functions if the functions are $f\left(x\right),g\left(x\right)$.

$\left(\raisebox{1ex}{$f$}\!\left/ \!\raisebox{-1ex}{$g$}\right.\right)x=\frac{f\left(x\right)}{g\left(x\right)}$

Therefore,

$\begin{array}{c}\left(\raisebox{1ex}{$f$}\!\left/ \!\raisebox{-1ex}{$g$}\right.\right)x=\frac{f\left(x\right)}{g\left(x\right)}\\ \left(\raisebox{1ex}{$f$}\!\left/ \!\raisebox{-1ex}{$g$}\right.\right)x=\frac{x^2+3}{x^2-3}\end{array}$

Hence, the required expression for the provided functions $f\left(x\right)=x^2+3,g\left(x\right)=x^2-3$ is $\left(\raisebox{1ex}{$f$}\!\left/ \!\raisebox{-1ex}{$g$}\right.\right)x=\frac{x^2+3}{x^2-3}$.

To determine

To calculate: The expression which represents $\left(f\cdot f\right)\left(x\right)$ for the function, $f\left(x\right)=x^2+3.$

Answer to Problem 1DE

Solution:

The required expression for the provided function $f\left(x\right)=x^2+3.$ $\left(f\cdot f\right)\left(x\right)={\left(x^2+3\right)}^2$.

Explanation of Solution

Given information:

A function,

$f\left(x\right)=x^2+3$

Formula used:

Multiplication of a function with itself.

If there a function,

Then,

$\left(f\cdot f\right)\left(x\right)=f\left(x\right)\cdot f\left(x\right)$

Calculation:

Consider the function provided.

$f\left(x\right)=x^2+3$

Use the principal of multiplication of function with itself.

$\left(f\cdot f\right)\left(x\right)=f\left(x\right)\cdot f\left(x\right)$

Therefore,

$\begin{array}{c}\left(f\cdot f\right)\left(x\right)=f\left(x\right)\cdot f\left(x\right)\\ =\left(x^2+3\right)\left(x^2+3\right)\\ \left(f\cdot f\right)\left(x\right)={\left(x^2+3\right)}^2\end{array}$

Hence, the expression that represents the required function is $\left(f\cdot f\right)\left(x\right)={\left(x^2+3\right)}^2$.