Find the domain of each function. f(x) = 1/[4/(x - 1) - 2]

검증된 단계별 안내

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. Identify the denominator of the given function: \( \frac{3}{x - 1} - 2 \).

Step 2: Set the denominator equal to zero to find the x-values that make the function undefined: \( \frac{3}{x - 1} - 2 = 0 \).

Step 3: Solve the equation \( \frac{3}{x - 1} - 2 = 0 \). First, isolate the fraction by adding 2 to both sides: \( \frac{3}{x - 1} = 2 \).

Step 4: Eliminate the fraction by multiplying both sides of the equation by \( x - 1 \) (note: \( x - 1 \neq 0 \) to avoid division by zero): \( 3 = 2(x - 1) \). Expand and solve for x: \( 3 = 2x - 2 \), then \( 2x = 5 \), and finally \( x = \frac{5}{2} \).

Step 5: Combine the results. The function is undefined when \( x = 1 \) (from the denominator \( x - 1 \neq 0 \)) and when \( x = \frac{5}{2} \) (from solving \( \frac{3}{x - 1} - 2 = 0 \)). Therefore, the domain of the function is all real numbers except \( x = 1 \) and \( x = \frac{5}{2} \).

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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 typically restricted by values that would make the denominator zero, as division by zero is undefined.

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/[3/(x - 1) - 2], the denominator must be analyzed to ensure it does not equal zero, which is crucial for determining the domain.

Solving for Undefined Points

To find the domain, one must identify values of x that make the denominator zero. This involves setting the denominator equal to zero and solving for x. Any x-value that results in a zero denominator must be excluded from the domain.