Chapter 2.6, Problem 32E

Solution

The match of the functions, $f\left(x\right)=-\frac{1}{x+3}$ with given graphs.

The graph of $f\left(x\right)=-\frac{1}{x+3}$ is (b).

Given:

The graphs are given as follows,

  

  

Concept Used:

The graph of a function of type $f\left(x\right)=\frac{ax+b}{cx+d}$ where $c\ne 0$ can be obtained by using the transformation of basic reciprocal function $\frac{1}{x}$ .

The graph of basic reciprocal function $f\left(x\right)=\frac{1}{x}$ is given by,

  

Here, horizontal asymptotes is $x=0$ , vertical asymptotes is, $y=0$ and $\underset{x\to {\infty }^{-}}{\lim }f\left(x\right)=\underset{x\to {\infty }^{+}}{\lim }f\left(x\right)=0$ .

And, by reflection property, the of graph $-f\left(x\right)$ is the reflection of $f\left(x\right)$ on x -axis.

Calculation:

Consider the function, $f\left(x\right)=-\frac{1}{x+3}$ .

If the basic reciprocal function is $g\left(x\right)=\frac{1}{x}$ , then $f\left(x\right)=-g\left(x+3\right)$ is the reflection of $g\left(x-3\right)$ on x- axis which is a translation of three units left of graph of $g\left(x\right)=\frac{1}{x}$ .

Hence, the graph of $f\left(x\right)=-\frac{1}{x+3}$ is reflection and translation of basic function $g\left(x\right)=\frac{1}{x}$ . The graph of $f\left(x\right)=-\frac{1}{x+3}$ is as follows,

  

Hence, from the given graphs, the graph of $f\left(x\right)=-\frac{1}{x+3}$ is (b).

Conclusion:

The graph of $f\left(x\right)=-\frac{1}{x+3}$ is (b).