In Exercises 11–26, determine whether each equation defines y as a function of x. xy - 5y =1

검증된 단계별 안내

Start with the given equation: \(xy - 5y = 1\).

Factor out \(y\) from the left side: \(y(x - 5) = 1\).

Solve for \(y\) by dividing both sides by \((x - 5)\), assuming \(x \neq 5\): \(y = \frac{1}{x - 5}\).

Analyze the expression for \(y\): for each value of \(x\) (except \(x = 5\)), there is exactly one corresponding value of \(y\).

Conclude that since \(y\) can be written as a single-valued function of \(x\) (except at \(x = 5\) where it is undefined), the equation defines \(y\) as a function of \(x\) on its domain.

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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 an equation defines y as a function of x, we check if for every x there is only one y that satisfies the equation.

Implicit vs. Explicit Functions

An explicit function expresses y directly in terms of x (e.g., y = f(x)), while an implicit function involves both variables in an equation (e.g., xy - 5y = 1). Understanding how to manipulate implicit equations helps determine if y can be uniquely solved for each x.

Solving for y and the Vertical Line Test

To check if y is a function of x, solve the equation for y. If you get one unique y for each x, it is a function. Graphically, the vertical line test states that if any vertical line crosses the graph more than once, y is not a function of x.