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

step 1 So we're given this function. And the first thing is we need to find out what makes the denomina

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

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

Key Concepts

Slant (Oblique) Asymptotes

Slant or oblique asymptotes occur when the degree of the numerator is exactly one more than the degree of the denominator in a rational function; the function will then follow a linear trend as x approaches infinity or minus infinity. Identifying slant asymptotes involves polynomial long division and is important for accurately depicting the long-term behavior of the function.

Graph Behavior Near Asymptotes

Understanding the behavior of a rational function near its asymptotes involves analyzing how the function approaches vertical, horizontal, or slant asymptotes. This includes noting the direction of divergence near vertical asymptotes and understanding the convergence towards horizontal or slant asymptotes, which helps in constructing an accurate graph of the function.

Removable Discontinuities (Holes)

A removable discontinuity, or hole, in a rational function happens when a factor is common to both the numerator and the denominator and cancels out, leading to a point where the function is not defined even though the overall behavior is consistent with the neighboring points. Recognizing holes is important to precisely sketch the function’s graph and understand its limits.

Domain of Rational Functions

The domain of a rational function includes all real numbers except where the denominator is equal to zero. This concept is critical because it determines the valid input values that do not lead to undefined expressions, such as division by zero, and is derived by setting the denominator not equal to zero and solving the resulting equation.

Vertical Asymptotes

Vertical asymptotes occur in rational functions when the denominator approaches zero while the numerator is non-zero at the same point, resulting in the function values tending to positive or negative infinity. This concept is essential for understanding discontinuities in the graph of a function and the behavior nearby these lines.

Horizontal Asymptotes

Horizontal asymptotes describe the behavior of a function as the input values tend to positive or negative infinity and are determined by comparing the degrees of the numerator and denominator in rational functions. This concept helps predict the end behavior of the function and its stabilization level for large values of x.