Provide a short answer to each question. What is the domain of the function ƒ(x)=1/x? What is its range?

검증된 단계별 안내

Identify the function given: \(f(x) = \frac{1}{x}\). This is a rational function where the denominator cannot be zero.

Determine the domain by finding all values of \(x\) for which the function is defined. Since division by zero is undefined, exclude \(x = 0\) from the domain.

Express the domain in set notation: all real numbers except zero, which can be written as \(\{ x \in \mathbb{R} \mid x \neq 0 \}\).

To find the range, consider the possible output values of \(f(x)\). Since \(f(x) = \frac{1}{x}\), as \(x\) approaches zero from either side, \(f(x)\) grows without bound positively or negatively, and as \(x\) becomes very large or very small, \(f(x)\) approaches zero but never equals zero.

Conclude that the range is all real numbers except zero, expressed as \(\{ y \in \mathbb{R} \mid y \neq 0 \}\).

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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. For ƒ(x) = 1/x, the function is undefined when the denominator is zero, so x cannot be zero. Therefore, the domain includes all real numbers except zero.

Range of a Function

The range of a function is the set of all possible output values (y-values) that the function can produce. For ƒ(x) = 1/x, the output can be any real number except zero, because 1/x never equals zero for any real x. Thus, the range is all real numbers except zero.

Rational Functions and Undefined Points

A rational function is a ratio of two polynomials, and it is undefined where the denominator equals zero. In ƒ(x) = 1/x, the denominator x cannot be zero, which creates a vertical asymptote at x = 0. Understanding this helps identify domain restrictions and behavior near undefined points.