Chapter 4.4, Problem 41AYU

In Problems 33-44, (a) graph the rational function using transformations, (b) use the final graph to find the domain and range, and (c) use the final graph to list any vertical, horizontal, or oblique asymptotes.

$G\left(x\right)\text{ = 1 + }\frac{2}{{(x-\text{ 3)}}^2}$

To determine

To find:

a.Graph the rational function using transformations,

Answer to Problem 41AYU

a.$y=\frac{1}{ x^2 }$

$y=\frac{1}{ { \left( x-3 \right)}^2 }$

(Replace $x$ by $x-3$; shift right 3 units)

$y=\frac{2}{ { \left( x-3 \right)}^2 }$

(vertical stretch about the $x\text{-axis}$ at 2)

$y=\frac{2}{ { \left( x-3 \right)}^2 }+1$

(Vertical translation up 1)

Explanation of Solution

Given:

$G\left( x \right)=1+\frac{2}{ { \left( x-3 \right)}^2 }$

To determine

To find:

b.Use the final graph to find the domain and range and

Answer to Problem 41AYU

b.The domain is $\left\{ x|x>1\right\}$ and range is $\left\{ y| y\ne 3\right\}$.

Explanation of Solution

Given:

$G\left( x \right)=1+\frac{2}{ { \left( x-3 \right)}^2 }$

To determine

To find:

c.Use the final graph to list any vertical, horizontal or oblique asymptotes.

Answer to Problem 41AYU

c.Horizontal asymptotes: $y=3$.

Vertical asymptotes $x=1$.

Oblique asymptotes: None.

Explanation of Solution

Given:

$G\left( x \right)=1+\frac{2}{ { \left( x-3 \right)}^2 }$