Graph each rational function. See Examples 5–9.
$\text{ƒ}\left(x\right)=(16x^2-9)/(x^2-9)$

Verified step by step guidance

1. Identify the rational function given: \(f(x) = \frac{16x^2 - 9}{x^2 - 9}\).
2. Factor both the numerator and the denominator to simplify and find key features: factor \$16x^2 - 9\( as a difference of squares and \)x^2 - 9$ as well.
3. Determine the domain by finding values of \(x\) that make the denominator zero, since these values are excluded from the domain and may indicate vertical asymptotes or holes.
4. Find the vertical asymptotes by setting the denominator equal to zero and solving for \(x\), then check if any factors cancel with the numerator to identify holes instead of asymptotes.
5. Find the horizontal asymptote by comparing the degrees of the numerator and denominator polynomials and using the rules for end behavior of rational functions.

Key Concepts

Rational Functions and Their Domains

A rational function is a ratio of two polynomials. Its domain includes all real numbers except where the denominator equals zero, as division by zero is undefined. Identifying these values helps determine vertical asymptotes and restrictions on the graph.

Asymptotes of Rational Functions

Asymptotes are lines that the graph approaches but never touches. Vertical asymptotes occur where the denominator is zero, and horizontal or oblique asymptotes describe end behavior based on the degrees of numerator and denominator polynomials.

Graphing Techniques for Rational Functions

Graphing involves finding intercepts, asymptotes, and analyzing behavior near undefined points. Simplifying the function, plotting key points, and understanding limits near asymptotes help create an accurate sketch of the rational function.