In Exercises 89–94, the equation for f is given by the simplified expression that results after performing the indicated operation. Write the equation for f and then graph the function. x/(2x+6) − 9/(x²−9)

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Identify the given expression:

\frac{x}{2 x + 6} - \frac{9}{x^{2}}

Factor the denominators where possible:

2 x + 6 = 2 (x + 3)

and

x^{2} - 9 = (x - 3)(x + 3)

Find the least common denominator (LCD) for the two fractions, which is

2(x - 3)(x + 3)

Rewrite each fraction with the LCD as the denominator by multiplying numerator and denominator appropriately: multiply the first fraction by

(x - 3)/(x - 3)

and the second fraction by

2/2

Combine the two fractions into a single fraction by subtracting the numerators over the common denominator, then simplify the numerator algebraically.

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

영상 길이:

17m

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Simplifying Rational Expressions

Simplifying rational expressions involves factoring polynomials in the numerator and denominator and reducing common factors. This process makes complex fractions easier to work with and is essential before performing operations like subtraction or addition.

Operations with Rational Expressions

To subtract rational expressions, you must find a common denominator, rewrite each fraction with this denominator, and then combine the numerators. Understanding how to manipulate denominators and numerators correctly is crucial for accurate simplification.

Graphing Rational Functions

Graphing rational functions requires identifying key features such as domain restrictions, vertical and horizontal asymptotes, and intercepts. These features help visualize the behavior of the function and are determined from the simplified expression.

In Exercises 89–94, the equation for f is given by the simplified expression that results after performing the indicated operation. Write the equation for f and then graph the function. x/(2x+6) − 9/(x2−9)