5. Rational Functions

Find the domain of each rational function. g(x)=3x²/(x−5)(x+4)

Give the equations of any vertical, horizontal, or oblique asymptotes for the graph of each rational function. ƒ(x)=(x²-2x-3)/(2x²-x-10)

Use the graphs of the rational functions in choices A–D to answer each question.

There may be more than one correct choice. Which choices have domain (-∞, 3)U(3, ∞)?

Find the vertical asymptotes, if any, and the values of x corresponding to holes, if any, of the graph of each rational function. h(x)=(x+7)/(x²+4x−21)

Solve each problem. Find a rational function ƒ having the graph shown.

Work each problem. Choices A–D below show the four ways in which the graph of a rational function can approach the vertical line x=2 as an asymptote. Identify the graph of each rational function defined in parts (a) – (d). $\text{ƒ}\left(x\right)=-1/(x-2{)}^2$

Use the graph of the rational function in the figure shown to complete each statement in Exercises 9–14.

As $x\to -\infty ,f\left(x\right){\to }_{{}_{}}$_____

Graph each rational function. ƒ(x)=3x/(x²-x-2)

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

In Exercises 89–94, the equation for f is given by the simplified expression that results after performing the indicated operation. Write the equation for f and then graph the function. (1 − 3/(x+2)) / (1 + 1/(x−2))

Sketch the graph of the function $f\(\left\)(x\(\right\))=\(\frac{1}{x^2}\)$. Identify the asymptotes on the graph.

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

Match the rational function in Column I with the appropriate description in Column II. Choices in Column II can be used only once.

$\text{ƒ}\left(x\right)=\frac{(x+7)}{(x+1)}$

Use the graph of the rational function in the figure shown to complete each statement in Exercises 9–14.

As $x\to 1^{-},f\left(x\right){\to }_{{}_{}}$______

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