교과서 질문
Find the inverse of f(x) = x3 + 2
685
views
Ch. 3 - Polynomial and Rational Functions
Blitzer8th EditionCollege AlgebraISBN: 9780136970514
당신이 사용하는 게 아니라요?교과서 변경다른 교과서를 선택하세요
TOC.tableOfContents
- Ch. P - Fundamental Concepts of Algebra311
- Ch. 1 - Equations and Inequalities398
- Ch. 2 - Functions and Graphs407
- Ch. 3 - Polynomial and Rational Functions306
- Ch. 4 - Exponential and Logarithmic Functions247
- Ch. 5 - Systems of Equations and Inequalities173
- Ch. 6 - Matrices and Determinants143
- Ch. 7 - Conic Sections124
- Ch. 8 - Sequences, Induction, and Probability251
4장, 문제 61
Follow the seven steps to graph each rational function. f(x)=2x2/(x2−1)
검증된 단계별 안내1
Identify the domain of the function by finding the values of \(x\) that make the denominator zero. Solve the equation \(x^{2} - 1 = 0\) to find these values, since the function is undefined there.
Find the vertical asymptotes by setting the denominator equal to zero and solving for \(x\). These are the lines where the function approaches infinity or negative infinity.
Determine the horizontal or oblique asymptote by comparing the degrees of the numerator and denominator. Since both numerator and denominator are degree 2, find the horizontal asymptote by dividing the leading coefficients.
Find the intercepts: For the \(y\)-intercept, evaluate \(f(0)\). For the \(x\)-intercepts, set the numerator equal to zero and solve for \(x\).
Analyze the behavior of the function near the vertical asymptotes and at points in each interval determined by the vertical asymptotes to understand how the graph behaves on those intervals.
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 points and to analyze behavior such as vertical asymptotes.
추천 영상:
6:04
Intro to Rational Functions
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 as x approaches infinity or negative infinity.
추천 영상:
6:24Introduction to Asymptotes
Graphing Steps for Rational Functions
Graphing involves identifying domain restrictions, intercepts, asymptotes, and behavior near asymptotes, then plotting points to sketch the curve. Following a systematic seven-step process ensures a complete and accurate graph.
추천 영상:
8:19How to Graph Rational Functions
관련 실천
교과서 질문
Find the domain of each function.
589
views
교과서 질문
In Exercises 57–64, find the vertical asymptotes, if any, the horizontal asymptote, if one exists, and the slant asymptote, if there is one, of the graph of each rational function. Then graph the rational function. r(x) = (x^2 + 4x + 3)/(x + 2)^2
693
views
교과서 질문
Among all pairs of numbers whose sum is 16, find a pair whose product is as large as possible. What is the maximum product?
1351
views
교과서 질문
Find the domain of each function.
523
views
교과서 질문
Find the domain of each function.
521
views