5. Rational Functions

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

Use transformations of f(x)=1/x or f(x)=1/x² to graph each rational function. g(x)=1/(x−1)

Use transformations of f(x)=1/x or f(x)=1/x² to graph each rational function. h(x)=1/(x−3)²+1

Follow the seven steps to graph each rational function. f(x)=4x/(x−2)

Follow the seven steps to graph each rational function. f(x)=2/(x²+x−2)

Follow the seven steps to graph each rational function. f(x)=2x²/(x²+4)

In Exercises 81–88, a. Find the slant asymptote of the graph of each rational function and b. Follow the seven-step strategy and use the slant asymptote to graph each rational function. f(x)=(x²−1)/x

In Exercises 81–88, a. Find the slant asymptote of the graph of each rational function and b. Follow the seven-step strategy and use the slant asymptote to graph each rational function. f(x)=(x²+1)/x

In Exercises 81–88, a. Find the slant asymptote of the graph of each rational function and b. Follow the seven-step strategy and use the slant asymptote to graph each rational function. f(x)=(x³+1)/(x²+2x)