Combining Linear Functions

We want to perform arithmetic operations with the functions $f(x)$ and $g(x).$

Calculating $f(x)+g(x)$

Let's substitute the given function rules into the expressions $f(x)$ and $g(x).$ Then we will perform the addition. After we have performed the addition, the result takes the form of $3x-9.$ $$\begin{array}{c}f(x)+g(x)=3x-9\end{array}$$

Calculating $f(x)-g(x)$

Let's substitute the given function rules into the expressions $f(x)$ and $g(x).$ Then we will perform the subtraction. After we have performed the subtraction, the result takes the form of $\text{-}x+5.$ $$\begin{array}{c}f(x)-g(x)=\text{-}x+5\end{array}$$

Calculating $f(x)\cdot g(x)$

Let's substitute the given function rules into the expressions $f(x)$ and $g(x).$ Then we will perform the multiplication. After we have performed the multiplication, the result takes the form of $2x^2-11x+14.$ $$\begin{array}{c}f(x)\cdot g(x)=2x^2-11x+14\end{array}$$

Calculating $\frac{f(x)}{g(x)}$

To perform the division, let's substitute the given function rules into the expressions $f(x)$ and $g(x).$ This expression cannot be simplified any further. The domain of both $f(x)$ and $g(x)$ is all real numbers, $\mathbb{R}.$ The domain of $\frac{f(x)}{g(x)}$ is the set of numbers common to the domains of both $f(x)$ and $g(x).$ However, the denominator cannot be $0.$ Let's find the values that make $g(x)=0.$ As we can see, $x=\frac{7}{2}$ makes $g(x)=0.$ Therefore, this value must be excluded from the domain of the quotient. We can say that the domain is the set of all real numbers except $\frac{7}{2}.$ $$\begin{array}{c}\frac{f(x)}{g(x)}=\frac{x-2}{2x-7},x\ne \frac{7}{2}\end{array}$$