Graph each rational function. See Examples 5–9.ƒ(x)=(16x2−9)/(x2−9)ƒ(x)=$$(16x^2$-9)/(x^2-9)$$

Ch. 3 - Polynomial and Rational Functions

Lial13th EditionCollege AlgebraISBN: 9780136881063당신이 사용하는 게 아니라요?교과서 변경

모든 교과서

Lial 13th Edition

Ch. 3 - Polynomial and Rational Functions

문제 76

4장, 문제 76

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

검증된 단계별 안내

Identify the rational function given: $f(x) = \frac{16x^2 - 9}{x^2 - 9}$.

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.

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.

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.

Find the horizontal asymptote by comparing the degrees of the numerator and denominator polynomials and using the rules for end behavior of rational functions.

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

영상 길이:

17m

도움이 되었나요?

주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

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.

Graph each rational function. See Examples 5–9.ƒ(x)=(16x2−9)/(x2−9)ƒ(x)=(16x^2-9)/(x^2-9)