Find the horizontal asymptote, if there is one, of the graph of each rational function. f(x)=12x/(3x²+1)

검증된 단계별 안내

Identify the degrees of the numerator and denominator polynomials in the rational function $f(x) = \frac{12x}{3x^{2} + 1}$.

Recall that the degree of the numerator is 1 (since \$12x$ is a first-degree polynomial) and the degree of the denominator is 2 (since \$3x^{2}$ is a second-degree polynomial).

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 $y = 0$.

Since the degree of the numerator (1) is less than the degree of the denominator (2), conclude that the horizontal asymptote is $y = 0$.

Write the final conclusion that the graph of $f(x)$ has a horizontal asymptote at $y = 0$.

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

영상 길이:

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 the behavior of rational functions involves analyzing the degrees and coefficients of the numerator and denominator polynomials.

Horizontal Asymptotes

A horizontal asymptote describes the behavior of a function as x approaches infinity or negative infinity. It is a horizontal line y = L that the graph approaches but does not necessarily touch, indicating the end behavior of the function.

Degree Comparison of Polynomials

To find horizontal asymptotes of rational functions, compare the degrees of the numerator and denominator polynomials. If the numerator's degree is less, the asymptote is y=0; if equal, it is the ratio of leading coefficients; if greater, there is no horizontal asymptote.

Find the horizontal asymptote, if there is one, of the graph of each rational function. f(x)=12x/(3x2+1)