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

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Identify the function given: \(g(x) = \frac{3}{x-4}\). Since this is a rational function, the domain includes all real numbers except where the denominator is zero.

Set the denominator equal to zero to find values that are not allowed: \(x - 4 = 0\).

Solve the equation for \(x\): \(x = 4\). This value makes the denominator zero, which is undefined in mathematics.

Therefore, the domain of \(g(x)\) is all real numbers except \(x = 4\).

Express the domain in interval notation: \((-\infty, 4) \cup (4, \infty)\).

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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 is the set of all possible input values (x-values) for which the function is defined. It excludes any values that cause undefined expressions, such as division by zero or taking the square root of a negative number in real numbers.

Restrictions from Denominators

When a function includes a denominator, the values that make the denominator zero are excluded from the domain because division by zero is undefined. To find these restrictions, set the denominator equal to zero and solve for x.

Rational Functions

A rational function is a ratio of two polynomials. Its domain includes all real numbers except those that make the denominator zero. Understanding rational functions helps in identifying domain restrictions and behavior near excluded values.