5. Rational Functions

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). $\text{ƒ}\left(x\right)=-1/(x-2)$

In Exercises 21–36, find the vertical asymptotes, if any, and the values of x corresponding to holes, if any, of the graph of each rational function. r(x)=x/(x^2+4)

Graph the rational function using transformations.

$f\left(x\right)=-\frac{1}{x}+3$

Graph the rational function using transformations.

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

Graph the rational function.

$f\left(x\right)=\frac{x+3}{x^2+5x+6}$

Graph each rational function. See Examples 5–9. ƒ(x)=[(x-5)(x-2)]/(x^2+9)

Graph each rational function. See Examples 5–9. ƒ(x)=(3x^2+3x-6)/(x^2-x-12)

Graph each rational function. See Examples 5–9.

$\text{ƒ}\left(x\right)=x/(4-x^2)$

Graph each rational function. See Examples 5–9. ƒ(x)=(x+2)/(x-3)

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?

In Exercises 57–64, find the vertical asymptotes, if any, the horizontal asymptote, if one exists, and the slant asymptote, if there is one, of the graph of each rational function. Then graph the rational function. g(x) = (2x - 4)/(x + 3)

In Exercises 57–64, find the vertical asymptotes, if any, the horizontal asymptote, if one exists, and the slant asymptote, if there is one, of the graph of each rational function. Then graph the rational function. h(x) = (x^2 - 3x - 4)/(x^2 - x -6)

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

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In Exercises 37–44, find the horizontal asymptote, if there is one, of the graph of each rational function. f(x)=(−2x+1)/(3x+5)

In Exercises 37–44, find the horizontal asymptote, if there is one, of the graph of each rational function. h(x)=12x³/(3x²+1)