Ch. 3 - Polynomial and Rational Functions

Blitzer8th EditionCollege AlgebraISBN: 9780136970514당신이 사용하는 게 아니라요?교과서 변경

Blitzer 8th Edition

Ch. 3 - Polynomial and Rational Functions

문제 79

4장, 문제 79

In Exercises 57–80, follow the seven steps to graph each rational function. f(x)=(3x²+x−4)/(2x²−5x)

검증된 단계별 안내

Identify the domain of the function by finding the values of

x

that make the denominator zero. Solve

2x^2 - 5x = 0

to find these values, since the function is undefined there.

Find the intercepts: For the y-intercept, evaluate

f(0)

if it is in the domain. For the x-intercepts, set the numerator equal to zero and solve

3x^2 + x - 4 = 0

to find the values of

x

where

f(x) = 0

Determine any vertical asymptotes by using the values of

x

that make the denominator zero (from step 1), provided these values do not also make the numerator zero (which would indicate a hole instead).

Find the horizontal or oblique (slant) asymptote by comparing the degrees of the numerator and denominator polynomials. Since both are degree 2, divide the leading coefficients to find the horizontal asymptote.

Analyze the behavior of the function near the vertical asymptotes and at the ends of the domain by testing values of

x

in each interval determined by the vertical asymptotes, and use this information to sketch the graph.

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

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 to avoid undefined values and to analyze the function's behavior.

Asymptotes

Asymptotes are lines that the graph of a function approaches but never touches. Vertical asymptotes occur where the denominator is zero, and horizontal or oblique asymptotes describe end behavior as x approaches infinity or negative infinity.

Graphing Steps for Rational Functions

Graphing rational functions involves identifying domain restrictions, intercepts, asymptotes, and behavior near asymptotes, then plotting points to sketch the curve. Following a systematic approach ensures an accurate and complete graph.

In Exercises 57–80, follow the seven steps to graph each rational function. f(x)=(3x2+x−4)/(2x2−5x)