Chapter 2.2, Problem 28AYU

In Problems 25-30, answer the questions about the given function.

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

(a) Is the point $\left(\frac{1}{2},-\frac{2}{3}\right)$ 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 $f\left(x\right)=1$, what is $x$? What point(s) are on the graph of $f$?

(d) What is the domain of $f$?

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

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

To determine

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

a. Is the point $(\frac{1}{2}, -\frac{2}{3})$ on the graph of $f?$

Answer to Problem 28AYU

Solution:

a. Yes, the point $(\frac{1}{2}, -\frac{2}{3})$ is on the graph of $f$.

Explanation of Solution

Given:

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

Calculation:

a. $f\left(x\right) = \frac{2x}{x - 2}$

To find the point $(\frac{1}{2}, -\frac{2}{3})$ on the graph of $f$, substitute $x = \frac{1}{2}$ in $f\left(x\right)$

$f\frac{1}{2} = \frac{2*\frac{1}{2}}{\frac{1}{2} - 2}$

$= \frac{1}{\frac{1 - 4}{2}}$

$= -\frac{2}{3}$

Yes, the point $(\frac{1}{2}, -\frac{2}{3})$ is on the graph of $f$.

To determine

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

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

Answer to Problem 28AYU

Solution:

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

Explanation of Solution

Given:

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

Calculation:

b. If $x = 4$

$f\left(2\right) = \frac{2\left(4\right)}{4 - 2}$

$= \frac{8}{2}$

$=4$

The point $(4,\text{ 4)}$ is on the graph of $f$.

To determine

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

c. If $f\left(x\right)=1$ what is $x?$ What point(s) are on the graph of $f?$

Answer to Problem 28AYU

Solution:

c. $-2$; The points $\left(-2,1\right)$ is on the graph of $f$.

Explanation of Solution

Given:

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

Calculation:

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

$2x = 1\left(x - 2\right)$

$2x = x - 2$

$2x - x = - 2$

$x = -2$

The point $\left(1, -2\right)$ is on the graph.

To determine

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

d. What is the domain of $f?$

Answer to Problem 28AYU

Solution:

d. Domain of $f = \left\{x|x \ne 2\right\}$

Explanation of Solution

Given:

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

Calculation:

d. Domain of $f = \left\{x|x \ne 2\right\}$.

To determine

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

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

Answer to Problem 28AYU

Solution:

e. $x\text{- intercept = 0}$

Explanation of Solution

Given:

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

Calculation:

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

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

$2x =0$

$\Rightarrow x=0$

To determine

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

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

Answer to Problem 28AYU

Solution:

f. $y\text{-intercept =} 0$

Explanation of Solution

Given:

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

Calculation:

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

$f\left(0\right)= \frac{2\left(0\right)}{\left(0\right) - 2} = \frac{0}{1} = 0\hspace{0.17em}\Rightarrow \hspace{0.17em}y\text{-intercept} = 0$.