Question

Work each problem. Which function has a graph that does not have a horizontal asymptote? A. ƒ(x)=(2x-7)/(x+3) B. ƒ(x)=3x/(x2-9) C. ƒ(x)=(x2-9)/(x+3) D. ƒ(x)=(x+5)/(x+2)(x-3)

Accepted Answer

Recall that a horizontal asymptote describes the behavior of a function as $$x$$ approaches positive or negative infinity. It often depends on the degrees of the polynomials in the numerator and denominator of a rational function. For each function, identify the degree of the numerator and the degree of the denominator: A. $$f(x) = \frac{2x - 7}{x + 3}$$: numerator degree is 1, denominator degree is 1. B. $$f(x) = \frac{3x}{x^2$ - 9}$$: numerator degree is 1, denominator degree is 2. C. $$f(x) = \frac{x^2$ - 9}{x + 3}$$: numerator degree is 2, denominator degree is 1. D. $$f(x) = \frac{x + 5}{(x + 2)(x - 3)}$$: numerator degree is 1, denominator degree is 2 (since denominator expands to a quadratic). Use the rule for horizontal asymptotes of rational functions: If the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is $$y=0. If $$the degrees are equal, the horizontal asymptote is the ratio of the leading coefficients. If the degree of the numerator is greater than the degree of the denominator, there is no horizontal asymptote (there may be an oblique asymptote instead). Apply this rule to each function to determine which one does not have a horizontal asymptote.

Work each problem. Which function has a graph that does not have a horizontal asymptote?
A. ƒ(x)=(2x-7)/(x+3)
B. ƒ(x)=3x/(x²-9)
C. ƒ(x)=(x²-9)/(x+3)
D. ƒ(x)=(x+5)/(x+2)(x-3)

비슷한 문제에 대한 검증된 영상 답변:

주요 개념

Horizontal Asymptotes of Rational Functions

Degree of Polynomials in Rational Functions

Simplifying Rational Functions and Identifying Asymptotes