5. Rational Functions

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. ƒ(x)=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. ƒ(x)=(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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