5. Rational Functions

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

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

Identify any vertical, horizontal, or oblique asymptotes in the graph of $y=f\left(x\right)$. State the domain of $f$.

Give the equations of any vertical, horizontal, or oblique asymptotes for the graph of each rational function. ƒ(x)=(4x²+25)/(x²+9)

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). $\text{ƒ}\left(x\right)=-1/(x-2)$

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)

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

Graph each rational function. ƒ(x)=[(x-5)(x-2)]/(x²+9)

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

Graph each rational function. See Examples 5–9.

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

In Exercises 57–80, follow the seven steps to graph each rational function. f(x)=(x²+x−12)/(x²−4)

Solve each problem. This rational function has two holes and one vertical asymptote.

$\text{ƒ}\left(x\right)=\frac{(x^3+7x^2-25x-175)}{(x^3+3x^2-25x-75)}$

What are the x-values of the holes?

Give the equations of any vertical, horizontal, or oblique asymptotes for the graph of each rational function. ƒ(x)=(4-3x)/(2x+1)

Use long division to rewrite the equation for g in the form quotient + remainder/divisor. Then use this form of the function's equation and transformations of f(x) = 1/x to graph g. g(x)=(3x−7)/(x−2)

Find the horizontal asymptote, if there is one, of the graph of each rational function. h(x)=12x³/(3x²+1)