Chapter 9, Problem 38SGR

a.

To determine

To graph: The rational function $f\left(x\right)=\frac{10+x}{15+x}$ .

Answer to Problem 38SGR

The graph is drawn below.

Explanation of Solution

Given information:

The given rational function is $f\left(x\right)=\frac{10+x}{15+x}$ .

Calculation:

If $f\left(x\right)=\frac{a\left(x\right)}{b\left(x\right)}$ , $a\left(x\right)$ and $b\left(x\right)$ are polynomial function with no common factor other than 1, and $b\left(x\right)\ne 0$ , then $f\left(x\right)$ has vertical asymptotes whenever $b\left(x\right)=0$ .

Therefore, for the function $f\left(x\right)=\frac{10+x}{15+x}$ .

  $\begin{array}{l}f\left(x\right)=\frac{10+x}{15+x}\\ 15+x=0\\ x=-15\end{array}$

So the function is undefined for $x=-15$

Therefore, $x=-15$ is the vertical asymptote. Also,

Since, the degree of $a\left(x\right)$ is equal to the degree of $b\left(x\right)$ .

Therefore, horizontal asymptotes is the line $y=\frac{1}{1}=1$ .

Also,

  $\begin{array}{l}10+x=0\\ x=-10\end{array}$

There is a zero at $x=-10$ .

The graph of the function is drawn below:

  

b.

To determine

To find: What domain and range value are meaningful in the context of the problem.

Answer to Problem 38SGR

Domain is $x\ge 0$ and relevant range is $0<y<1$ .

Explanation of Solution

Given information:

The given rational function is $f\left(x\right)=\frac{10+x}{15+x}$ .

Calculation:

Since the given context represent a future sales, therefore the domain should be $x\ge 0$ .

From the graph of the function in part(a) the horizontal asymptote is at $y=1$ , thus the relevant range will be $0<y<1$ .

Hence,

Domain is $x\ge 0$ and relevant range is $0<y<1$ .