5. Rational Functions

Graph each rational function. ƒ(x)=(20+6x-2x²)/(8+6x-2x²)

Graph each rational function. ƒ(x)=(18+6x-4x²)/(4+6x+2x²)

Provide a short answer to each question. What is the domain of the function $\text{ƒ}\left(x\right)=\frac{1}{x^2}$? What is its range?

Graph each rational function. ƒ(x)=(9x²-1)/(x²-4)

Graph each rational function. See Examples 5–9.

$\text{ƒ}\left(x\right)=(16x^2-9)/(x^2-9)$

Provide a short answer to each question. What is the equation of the vertical asymptote of the graph of y=[1/(x-3)]+2? Of the horizontal asymptote?

Work each problem. Which function has a graph that does not have a vertical asymptote?

A. ƒ(x)=1/(x²+2)

B. ƒ(x)=1/(x²-2)

C. ƒ(x)=3/x²

D. ƒ(x)=(2x+1)/(x-8)

Identify any vertical, horizontal, or oblique asymptotes in the graph of y=ƒ(x). State the domain of ƒ.

Find a rational function ƒ having a graph with the given features.

x-intercepts: (-1, 0) and (3, 0)

y-intercept: (0, -3)

vertical asymptote: x=1

horizontal asymptote: y=1

Find a rational function ƒ having a graph with the given features.

x-intercepts: (1, 0) and (3, 0)

y-intercept: none

vertical asymptotes: x=0 and x=2

horizontal asymptote: y=1

Solve each problem. Work each of the following. Sketch the graph of a function that does not intersect its horizontal asymptote y=1, has the line x=3 as a vertical asymptote, and has x-intercepts (2, 0) and (4, 0).

Provide a short answer to each question. Is ƒ(x)=1/x an even or an odd function? What symmetry does its graph exhibit?

Graph each rational function. ƒ(x)=4/(x-1)

Use the graph of the rational function in the figure shown to complete each statement in Exercises 15–20.

As $x\to \infty ,f\left(x\right)\to$_____