Q. 12 - Precalculus Enhanced with Graphing Utilities

Question

Find $f+g,f-g,f\cdot g$ and $\frac{f}{g}$ for each pair of functions. State the domain of each of these functions.

$$\begin{gathered} f\left(x\right)=2-x \\ g\left(x\right)=3x+1 \end{gathered}$$

Step-by-Step Solution

Answer

For  $$\begin{gathered} f\left(x\right)=2-x \\ g\left(x\right)=3x+1 \end{gathered}$$, we get:

$f+g=2x+3$ and its domain is the set of all real numbers .

$f-g=-4x+1$ and its domain is the set of all real numbers .

$f\cdot g=-3x^2+5x+2$ and its domain is the set of all real numbers.

$\frac{f}{g}=\frac{2-x}{3x+1}$ and ints domain is $\left\{x\right|x\ne -\frac{1}{3}\}$.

Step 1. Given information

We have been given:

$$\begin{gathered} f\left(x\right)=2-x \\ g\left(x\right)=3x+1 \end{gathered}$$

We have to find $f+g,f-g,f\cdot g$ and $\frac{f}{g}$ for the pair of functions and state the domain of each of these functions.

Step 2. Find f + g and its domain.

We know $f+g=f\left(x\right)+g\left(x\right)$.

Therefore,

$$\begin{gathered} f+g \\ =(2-x)+(3x+1) \\ =2-x+3x+1 \\ =2x+3 \end{gathered}$$

Its domain is the set of all real numbers.

Step 3. Find f - g and its domain.

We know $f-g=f\left(x\right)+g\left(x\right)$.

$$\begin{gathered} f-g \\ =(2-x)-(3x+1) \\ =2-x-3x-1 \\ =1-4x \end{gathered}$$

Its domain is the set of all real numbers.

Step 4. Find f ⋅ g and its domain.

We know $f\cdot g=f\left(x\right)\cdot g\left(x\right)$.

$$\begin{gathered} f\cdot g \\ =(2-x)(3x+1) \\ =2(3x+1)-x(3x+1) \\ =6x+2-3x^2-x \\ =-3x^2+5x+2 \end{gathered}$$

Its domain is the set of all real numbers.

Step 5. Find f g and its domain.

We know $\frac{f}{g}=\frac{f\left(x\right)}{g\left(x\right)}$.

Therefore,

$$\begin{gathered} \frac{f}{g} \\ =\frac{2-x}{3x+1} \end{gathered}$$

Its domain will be the set of all real numbers except those which make the denominator zero.

$$\begin{gathered} 3x+1=0 \\ 3x=-1 \\ x=-\frac{1}{3} \end{gathered}$$

Domain is $\left\{x\right|x\ne -\frac{1}{3}\}$.