Determine whether each relation defines y as a function of x. Give the domain and range. See Example 5. y = 2x - 5

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Identify the given relation: \(y = 2x - 5\). This is an equation expressing \(y\) explicitly in terms of \(x\).

Determine if \(y\) is a function of \(x\): Since for every value of \(x\) there is exactly one corresponding value of \(y\) (because the equation is linear and passes the vertical line test), \(y\) is indeed a function of \(x\).

Find the domain: Since there are no restrictions on \(x\) in the equation \(y = 2x - 5\), the domain is all real numbers, which can be written as \((-\infty, \infty)\).

Find the range: Because \(y\) is a linear function with no restrictions, it can take any real value as \(x\) varies over all real numbers. Therefore, the range is also \((-\infty, \infty)\).

Summarize: The relation defines \(y\) as a function of \(x\) with domain \((-\infty, \infty)\) and range \((-\infty, \infty)\).

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Definition of a Function

A function is a relation where each input (x-value) corresponds to exactly one output (y-value). To determine if y is a function of x, check if for every x there is only one y. The given equation y = 2x - 5 is a linear function, so it passes this test.

Domain of a Function

The domain is the set of all possible input values (x-values) for which the function is defined. For the linear function y = 2x - 5, the domain is all real numbers because any real x can be substituted without restriction.

Range of a Function

The range is the set of all possible output values (y-values) that the function can produce. Since y = 2x - 5 is linear with no restrictions, its range is also all real numbers, as y can take any real value depending on x.