Ch. 3 - Polynomial and Rational Functions

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Ch. 3 - Polynomial and Rational Functions

문제 77

4장, 문제 77

In Exercises 57–80, follow the seven steps to graph each rational function. f(x)=(x²+x−12)/(x²−4)

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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, since division by zero is undefined.

Factor both the numerator and denominator to simplify the function if possible. Factor

x^2 + x - 12

and

x^2 - 4

to their binomial factors.

Determine the vertical asymptotes by setting the denominator equal to zero and solving for

x

. These are the values excluded from the domain where the function may approach infinity or negative infinity.

Find the horizontal or oblique asymptote by comparing the degrees of the numerator and denominator polynomials. Use the degree rules to determine the end behavior of the function.

Calculate the x-intercepts by setting the numerator equal to zero and solving for

x

, and find the y-intercept by evaluating

f(0)

if it is in the domain.

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주요 개념

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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.

Finding Asymptotes

Asymptotes are lines that the graph approaches but never touches. Vertical asymptotes occur where the denominator is zero (and 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 involves seven steps: determining domain, intercepts, asymptotes, sign analysis, and plotting points. This systematic approach helps visualize the function's shape and behavior accurately.

In Exercises 57–80, follow the seven steps to graph each rational function. f(x)=(x2+x−12)/(x2−4)