In Exercises 81–88, a. Find the slant asymptote of the graph of each rational function and b. Follow the seven-step strategy and use the slant asymptote to graph each rational function. f(x)=(x²−1)/x

검증된 단계별 안내

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

To find the slant (oblique) asymptote, perform polynomial long division of the numerator $x^{2} - 1$ by the denominator $x$.

Divide $x^{2}$ by $x$ to get $x$, then multiply $x$ by $x$ to get $x^{2}$, subtract this from $x^{2} - 1$ to find the remainder.

Next, divide the remainder by $x$ to find the next term of the quotient, continue until the degree of the remainder is less than the degree of the divisor.

The quotient (without the remainder) will be the equation of the slant asymptote, which you can write as $y = $ (quotient).

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

영상 길이:

16m

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), where Q(x) ≠ 0. Understanding the behavior of rational functions, including their domains and asymptotes, is essential for graphing and analyzing them.

Slant (Oblique) Asymptotes

Slant asymptotes occur when the degree of the numerator is exactly one more than the degree of the denominator. They represent the line that the graph approaches as x approaches infinity or negative infinity, found by performing polynomial division.

Polynomial Division

Polynomial division is a method used to divide one polynomial by another, similar to long division with numbers. It helps find the quotient and remainder, which are used to determine slant asymptotes and simplify rational functions for graphing.

In Exercises 81–88, a. Find the slant asymptote of the graph of each rational function and b. Follow the seven-step strategy and use the slant asymptote to graph each rational function. f(x)=(x2−1)/x