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

step 1 In this rational function, we have it graphed over here. You can use a graphing calculator. You

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. $$f(x)=\frac{3 x^{2}-5 x-2}{x^{2}-9}$$

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.
$$f(x)=\frac{3 x^{2}-5 x-2}{x^{2}-9}$$

Key Concepts

Slant (Oblique) Asymptotes

A slant or oblique asymptote occurs when the degree of the numerator is exactly one more than the degree of the denominator. In such cases, polynomial long division can be used to determine the equation of the slant asymptote, which describes the linear behavior of the function as x approaches infinity or negative infinity.

Behavior Near Asymptotes and Graphical Analysis

Graphical analysis of a rational function involves understanding how the function behaves near its vertical, horizontal, or slant asymptotes. This includes observing whether the function approaches positive or negative infinity near vertical asymptotes and how it stabilizes to a constant or linear function as x moves towards infinity. Visualization with graphing utilities helps in interpreting these behaviors and confirms the analytical results.

Horizontal Asymptotes

Horizontal asymptotes describe the end behavior of a rational function as x approaches positive or negative infinity. They are determined by comparing the degrees of the polynomial in the numerator and the denominator. Specifically, if the degrees are equal, the horizontal asymptote is given by the ratio of the leading coefficients.

Domain of Rational Functions

The domain of a rational function includes all real numbers except those that make the denominator zero. Determining the domain is crucial because it tells us the set of x-values for which the function is defined by solving for the values that make the denominator nonzero.

Vertical Asymptotes

Vertical asymptotes occur at the x-values where the denominator of a rational function becomes zero and the numerator is nonzero (after any simplification). They indicate points where the function grows without bound, and are identified by setting the simplified denominator equal to zero.

Removable Discontinuities (Holes)

Holes in the graph of a rational function occur when a factor in the numerator and denominator cancels, resulting in a common zero. Although the function is not defined at that specific x-value, the discontinuity is removable because it results from a common factor rather than an inherent characteristic of the rational expression.

Transcript

Please give Ace some feedback