Chapter 2.2, Problem 25AYU

To determine

To find: When $f\left(x\right)=\frac{x+2}{x-6}$, answer the following

a. Is the point $\left(3,14\right)$ on the graph of $f$?

b. If $x=4$, what is $f\left(x\right)$? What point is on the graph of $f$?

c. If $f\left(x\right)=2$, what is $x$? What point(s) are on the graph of $f$?

d. What is the domain of $f$?

e. List of $x\text{-intercepts}$, if any, of the graph of $f$.

f. List of the $y\text{-intercept}$, if there is one, of the graph of $f$.

Answer to Problem 25AYU

Solution:

a. No, the point $\left(3,14\right)$ is not on the graph of $f$.

b. $f\left(4\right)=-3$; The point $\left(4,-3\right)$ is on the graph of $f$.

c. 14; The points $\left(14,2\right)$ is on the graph of $f$.

d. Domain of $f=\left\{x|x\ne 6\right\}$.

e. $x=-2$.

f. $y\text{-intercept}=-\frac{1}{3}$.

Explanation of Solution

Given:

$f\left(x\right)=\frac{x+2}{x-6}$

Calculation:

1. $f\left(x\right)=\frac{x+2}{x-6}$

   To find the point $\left(3,14\right)$ on the graph of $f$, substitute $x=3$ in $f(x)$.

   $f\left(3\right)=\frac{3+2}{3-6}$

   $=\frac{5}{-3}$

   $=-\frac{5}{3}\ne 14$

   No, the point $\left(3,14\right)$ is not on the graph of $f$.

2. If $x=4$.

   $f(4)=\frac{4+2}{4-6}$

   $=\frac{6}{-2}$

   $=-3$

   The point $(4,-3)$ is on the graph of $f$.

3. $f\left(x\right)=2\Rightarrow \frac{x+2}{x-6} =2$

   $x+2=2\left(x-6\right)$

   $x+2=2x-12$

   $12+2=2x-x$

   $x=14$

   The points $\left(14,2\right)$ is on the graph of $f$.

4. Domain of $f=\left\{x|x\ne 6\right\}$.

5. To find $x\text{-intercept}$, let $f\left(x\right)=0$,

   $\frac{x+2}{x-6}=0$

   $x+2=0$

   $x=-2$

6. To find the $y\text{-intercept}$, let $x=0$ in $f\left(x\right)=\frac{x+2}{x-6}$.

   $f\left(0\right)=\frac{0+2}{0-6} =-\frac{1}{3}\Rightarrow y\text{-intercept}=-\frac{1}{3}$