Find the domain of each function. f(x) = 1/(x+7) + 3/(x-9)

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Step 1: Recall that the domain of a function is the set of all possible input values (x-values) for which the function is defined. For rational functions, the function is undefined when the denominator equals zero.

Step 2: Identify the denominators in the given function. The function is f(x) = 1/(x+7) + 3/(x-9). The denominators are (x+7) and (x-9).

Step 3: Set each denominator equal to zero to find the values of x that make the function undefined. Solve the equations x+7=0 and x-9=0.

Step 4: Solve x+7=0 to get x = -7, and solve x-9=0 to get x = 9. These are the values of x that make the denominators zero, so the function is undefined at x = -7 and x = 9.

Step 5: Write the domain of the function. The domain includes all real numbers except x = -7 and x = 9. In interval notation, the domain is expressed as (-∞, -7) ∪ (-7, 9) ∪ (9, ∞).

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주요 개념

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

Domain of a Function

The domain of a function refers to the set of all possible input values (x-values) for which the function is defined. For rational functions, the domain is restricted by values that make the denominator zero, as division by zero is undefined. Understanding the domain is crucial for determining where the function can be evaluated.

Rational Functions

A rational function is a function that can be expressed as the ratio of two polynomials. In the given function f(x) = 1/(x+7) + 3/(x-9), each term is a rational expression. The behavior of rational functions is significantly influenced by their denominators, which can introduce restrictions on the domain.

Finding Restrictions in the Domain

To find the domain of a function, one must identify values that cause the denominator to equal zero. For the function f(x) = 1/(x+7) + 3/(x-9), we set the denominators (x+7) and (x-9) to zero and solve for x. The solutions, x = -7 and x = 9, indicate the points where the function is undefined, thus defining the domain.