Graph each rational function. ƒ(x)=(9x2-1)/(x2-4)

Ch. 3 - Polynomial and Rational Functions

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Ch. 3 - Polynomial and Rational Functions

문제 75

4장, 문제 75

Graph each rational function. ƒ(x)=(9x²-1)/(x²-4)

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Identify the rational function given: \(f(x) = \frac{9x^2 - 1}{x^2 - 4}\).

Factor both the numerator and the denominator to simplify and find key features: factor \(9x^2 - 1\) as a difference of squares and \(x^2 - 4\) also as a difference of squares.

Determine the domain by finding values of \(x\) that make the denominator zero, since these values are excluded from the domain and correspond to vertical asymptotes or holes.

Find the vertical asymptotes by setting the denominator equal to zero and solving for \(x\), and check if any factors cancel with the numerator to identify holes instead.

Find the horizontal asymptote by comparing the degrees of the numerator and denominator polynomials, and analyze the end behavior of the function accordingly.

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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, zeros, and behavior depends on analyzing both numerator and denominator polynomials. Key features include vertical asymptotes where the denominator is zero and holes where factors cancel.

Asymptotes of Rational Functions

Asymptotes are lines that the graph approaches but never touches. Vertical asymptotes occur at values making the denominator zero (if not canceled), while horizontal or oblique asymptotes describe end behavior based on the degrees of numerator and denominator polynomials.

Factoring and Simplifying Polynomials

Factoring polynomials helps identify zeros and simplify the function. By factoring numerator and denominator, common factors can be canceled, revealing holes in the graph and simplifying the analysis of intercepts and asymptotes.