Q. 13 - 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.

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

Step-by-Step Solution

Answer

For  $f\left(x\right)=3x^2+x+1 ; g\left(x\right)=3x$, we get:

$f+g=3x^2+4x+1$ and its domain is the set of all real numbers.

$f-g=3x^2-2x+1$ and its domain is the set of all real numbers.

$f\cdot g=9x^3+3x^2+3x$ and its domain is the set of all real numbers.

$\frac{f}{g}=\frac{3x^2+x+1}{3x}$ and its domain is $\left\{x\right|x\ne 0\}$.

Step 1. Given information

We have been given:

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

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,

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

Its domain is the set of all real numbers. 

Step 3. Find f - g and its domain.

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

Therefore,

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

Its domain is the set of all real numbers. 

Step 4. Find f ⋅ g and its domain.

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

Therefore,

$$\begin{gathered} f\cdot g \\ =(3x^2+x+1)\left(3x\right) \\ =9x^3+3x^2+3x \end{gathered}$$

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,

$\frac{f}{g}=\frac{3x^2+x+1}{3x}$

The domain will be the set of all real numbers except those that make the denominator zero.

$$\begin{gathered} 3x=0 \\ x=0 \end{gathered}$$

The domain will be $\left\{x\right|x\ne 0\}$.