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

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

Key Concepts

Domain of Rational Functions

The domain of a rational function consists of all real numbers except where the denominator is zero. It is essential to identify the values that make the denominator vanish, as these are not included in the function’s domain.

Vertical Asymptotes

Vertical asymptotes occur at values where the function approaches infinity because the denominator is zero and the numerator is non-zero (after any cancellation). These asymptotes represent lines that the graph approaches but never crosses.

Removable Discontinuities

Removable discontinuities, or holes, occur when a factor in the denominator cancels with an identical factor in the numerator. Although the expression simplifies, the original function is not defined at the canceled value, resulting in a hole in the graph.

Horizontal Asymptote

Horizontal asymptotes describe the end behavior of a function when the degree of the numerator is equal to or less than the degree of the denominator. When the numerator’s degree exceeds that of the denominator, a horizontal asymptote does not exist.

Slant Asymptotes

Slant (or oblique) asymptotes occur when the degree of the numerator is exactly one more than that of the denominator. They are found by performing polynomial division, and the resulting quotient (ignoring the remainder) represents the slant asymptote.

Graph Behavior Near Asymptotes

Understanding the behavior of a function near its asymptotes is crucial in graph analysis. This includes examining how the function behaves as it approaches vertical asymptotes (typically diverging to positive or negative infinity) and how it follows the trend of horizontal or slant asymptotes at the far ends of the graph.

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