Find the domain of each function. h(x) = 4/(3/x - 1)

검증된 단계별 안내

Step 1: Recall that the domain of a function consists of all input values (x) for which the function is defined. For rational functions, the denominator cannot be zero because division by zero is undefined.

Step 2: Analyze the given function h(x) = 4/(3/x - 1). Notice that the denominator is (3/x - 1). To ensure the function is defined, set the denominator not equal to zero: 3/x - 1 ≠ 0.

Step 3: Solve the inequality 3/x - 1 ≠ 0. First, isolate the term 3/x by adding 1 to both sides: 3/x ≠ 1.

Step 4: Eliminate the fraction by multiplying both sides of the inequality by x (assuming x ≠ 0): 3 ≠ x. Additionally, x ≠ 0 must be considered because division by zero is undefined.

Step 5: Combine the restrictions: x ≠ 0 and x ≠ 3. The domain of h(x) is all real numbers except x = 0 and x = 3.

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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 case of h(x) = 4/(3/x - 1), the denominator is a polynomial expression, and identifying its zeros is crucial for determining the domain.

Finding Restrictions

To find the domain of a function, one must identify any restrictions on the variable. For the function h(x), we need to solve the equation in the denominator, 3/x - 1 = 0, to find values of x that would make the function undefined, thus determining the valid domain.