For the given rational function f : the domain of f. any vertical asymptotes of the graph of y=f

step 1 And this rational function where we're graphing it and identifying the characteristics of such,

For the given rational function f : the domain of f. any...

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. $$f(x)=\frac{x^{2}-x-12}{x^{2}+x-6}$$

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.
$$f(x)=\frac{x^{2}-x-12}{x^{2}+x-6}$$

Key Concepts

Graphing Rational Functions

Graphing rational functions involves combining knowledge of the domain, vertical asymptotes, holes, and horizontal or slant asymptotes. This integrated approach is used to sketch an accurate graph and to understand how the function behaves, especially near the asymptotes where the function’s values tend to infinity or drop sharply.

Slant Asymptotes

A slant asymptote, or oblique asymptote, occurs when the degree of the numerator is exactly one more than that of the denominator. It is found by performing polynomial long division and represents the line that the graph of the function will approach at extreme values of x.

Horizontal Asymptotes

Horizontal asymptotes describe the behavior of a rational function as x approaches positive or negative infinity. They are determined by comparing the degrees of the numerator and denominator. For instance, when the degrees are equal, the horizontal asymptote is the ratio of the leading coefficients, which provides insight into the end behavior of the function.

Vertical Asymptotes

Vertical asymptotes are vertical lines found at the x-values that make the denominator zero, provided these factors do not cancel with the numerator. They indicate where the function tends to infinity or negative infinity and are essential for understanding the behavior of the function near these discontinuities.

Domain of Rational Functions

The domain of a rational function consists of all real numbers except those that make the denominator zero. Understanding and determining the domain involves setting the denominator equal to zero and solving for the variable, which is a key step in analyzing any rational function.

Removable Discontinuities (Holes)

Holes occur in the graph of a rational function when a factor in the numerator and the denominator cancels out, leading to a point where the function is not defined even though the overall behavior of the function does not indicate an asymptote. Recognizing and factoring out these common factors is critical for correctly identifying the structure of the function.

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