For the given rational function f: ∙Find the domain of f. any vertical asymptotes of the graph o

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For the given rational function f: ∙Find the domain of f....

For the given rational function $f:$ $\bullet$ Find the domain of $f$. $\bullet$Identify any vertical asymptotes of the graph of $y=f(x)$. $\bullet$Identify any holes in the graph. $\bullet$Find the horizontal asymptote, if it exists. $\bullet$Find the slant asymptote, if it exists. $\bullet$Graph the function using a graphing utility and describe the behavior near the asymptotes. $f(x)=\frac{18-2 x^{2}}{x^{2}-9}$

For the given rational function $f:$
$\bullet$ Find the domain of $f$.
$\bullet$Identify any vertical asymptotes of the graph of $y=f(x)$.
$\bullet$Identify any holes in the graph.
$\bullet$Find the horizontal asymptote, if it exists.
$\bullet$Find the slant asymptote, if it exists.
$\bullet$Graph the function using a graphing utility and describe the behavior near the asymptotes.
$f(x)=\frac{18-2 x^{2}}{x^{2}-9}$

Key Concepts

Graph Behavior Near Asymptotes

The behavior of a rational function near its asymptotes is key to understanding its overall graph. Near vertical asymptotes, the function's values tend to approach infinity or negative infinity, while near horizontal or slant asymptotes, the function approaches these lines as the variable becomes very large in magnitude. This behavior provides insights into limits and the overall continuity of the function.

Rational Functions

A rational function is the ratio of two polynomials, and its behavior is largely determined by the degrees and factors of these polynomials. They are notable for their potential discontinuities and asymptotic behaviors, which help in understanding limits and continuity in calculus.

Domain of a Rational Function

The domain of a rational function includes all real numbers where the function is defined, which means excluding any values that make the denominator zero. Identifying these values is crucial for understanding the function's valid input range and overall behavior.

Vertical Asymptotes

Vertical asymptotes occur at values of the variable that make the denominator zero, provided that the zero does not cancel with a similar factor in the numerator. They indicate points where the function grows without bound, approaching positive or negative infinity, and signify a non-removable discontinuity.

Holes (Removable Discontinuities)

Holes arise in a rational function when a common factor is present in both the numerator and the denominator and cancels out. Although the simplified function appears to be defined everywhere, the original function is not defined at the specific point where the factor was zero, resulting in a removable discontinuity known as a hole.

Horizontal Asymptotes

Horizontal asymptotes describe the behavior of a rational function as the input variable tends toward infinity (positive or negative). They are determined by comparing the degrees of the numerator and denominator: if the degrees are equal, the horizontal asymptote is the ratio of the leading coefficients; if the numerator's degree is less, the asymptote is y=0.

Slant (Oblique) Asymptotes

Slant asymptotes occur when the degree of the numerator is exactly one higher than the degree of the denominator. They are found using polynomial long division, where the quotient (ignoring the remainder) represents the slant asymptote, indicating that the function will behave similarly to this linear expression for large absolute values of the variable.

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