Ch. 3 - Polynomial and Rational Functions

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Blitzer 8th Edition

Ch. 3 - Polynomial and Rational Functions

문제 73

4장, 문제 73

In Exercises 57–80, follow the seven steps to graph each rational function. f(x)=(x−2)/(x²−4)

검증된 단계별 안내

Identify the domain of the function by finding the values of

x

that make the denominator zero. Solve

x^2 - 4 = 0

to find these values.

Simplify the function if possible by factoring the numerator and denominator. Factor the denominator as

(x-2)(x+2)

and check for common factors with the numerator

(x-2)

Determine the vertical asymptotes by setting the denominator equal to zero and excluding any values canceled out during simplification. These are the values where the function is undefined and the graph may have vertical lines.

Find the horizontal or oblique asymptotes by comparing the degrees of the numerator and denominator polynomials. Use the rules for rational functions to determine the asymptote equation.

Calculate key points such as intercepts by setting

f(x) = 0

to find x-intercepts and evaluating

f(0)

to find the y-intercept. Plot these points along with the asymptotes to guide the sketch of the graph.

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

영상 길이:

15m

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주요 개념

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

Rational Functions

A rational function is a ratio of two polynomials, expressed as f(x) = P(x)/Q(x). Understanding its domain, where the denominator Q(x) ≠ 0, is essential because values that make the denominator zero are excluded and often correspond to vertical asymptotes or holes.

Asymptotes of Rational Functions

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

Graphing Steps for Rational Functions

Graphing rational functions involves identifying domain restrictions, intercepts, asymptotes, and behavior near asymptotes. Plotting key points and analyzing limits help sketch an accurate graph, following a systematic approach such as the seven-step method.

In Exercises 57–80, follow the seven steps to graph each rational function. f(x)=(x−2)/(x2−4)