Find the domain of each function. g(x) = 2/(x+5)

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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 denominator cannot be zero because division by zero is undefined.

Step 2: Identify the denominator of the given function. The function is g(x) = 2/(x+5), so the denominator is (x+5).

Step 3: Set the denominator equal to zero to find the x-value(s) that make the function undefined. Solve the equation x+5=0.

Step 4: Subtract 5 from both sides of the equation to isolate x. This gives x = -5. This is the value that makes the denominator zero, so it must be excluded from the domain.

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

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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. Understanding the domain is crucial because it determines the values that can be substituted into the function without resulting in undefined expressions.

Rational Functions

A rational function is a function that can be expressed as the ratio of two polynomials. In the case of g(x) = 2/(x+5), the denominator (x+5) must not equal zero, as division by zero is undefined. This restriction directly affects the domain of the function.

Finding Restrictions on the Domain

To find the domain of a function, one must identify any values that would make the function undefined. For g(x) = 2/(x+5), we set the denominator equal to zero (x+5=0) and solve for x, which reveals that x cannot equal -5. Thus, the domain excludes this value.