Chapter 7.4, Problem 107E

To determine

The graph of the given function and hence find its asymptotes.

Answer to Problem 107E

The given function has two asymptotes, a vertical asymptote x = -1 and a horizontal asymptote y = 0.

Explanation of Solution

Given: The given function is $f(x)=\frac{7}{-x-1}$ .

Concept Used: Consider any rational function $f(x)=\frac{g(x)}{h(x)}$ .

Vertical asymptotes of the function are given by $h(x)=0$

Horizontal asymptotes of the function are given by $y=\underset{x\to \infty }{{\displaystyle \lim }}f(x)$ .

Oblique asymptotes of the function are given by the linear part in the quotient after long division. Hence, if $f(x)=\frac{g(x)}{h(x)}=q(x)+\frac{r(x)}{h(x)}$ then the oblique asymptote is given by $y=q(x)$ . Note that oblique asymptotes of rational functions exists only when $\left|g(x)\right|>\left|h(x)\right|$ where $\left|F(x)\right|$ denotes the degree of $F(x)$ .

Graph: The graph of the given function is-

  

Interpretation: The given function is $f(x)=\frac{7}{-x-1}$ , which is a reciprocal function.

The points through which the given curve passes through are-

x	$f(x)$
0	-7
6	-1
-8	1
$\raisebox{1ex}{$-3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.$	14
$\raisebox{1ex}{$-1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.$	-14
$\raisebox{1ex}{$-9$}\!\left/ \!\raisebox{-1ex}{$2$}\right.$	2

Vertical asymptote of the function is given by

  $-x-1=0$

  $\Rightarrow x=-1$

Horizontal asymptote of the function is given by

  $y=\underset{x\to \infty }{{\displaystyle \lim }}f(x)$

  $\Rightarrow y=\underset{x\to \infty }{{\displaystyle \lim }}\frac{7}{-x-1}$

  $\Rightarrow y=0$

Since the degree of the function in the numerator is less than that of the denominator, hence the given function has no oblique asymptotes.

Thus the given function is a reciprocal function with two asymptotes, which are $x=-1$ and $y=0$ .