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

Verified step by step guidance

1. Start with the given equation: \(xy - 5y = 1\).
2. Factor out \(y\) from the left side: \(y(x - 5) = 1\).
3. Solve for \(y\) by dividing both sides by \((x - 5)\), assuming \(x \neq 5\): \(y = \frac{1}{x - 5}\).
4. Analyze the expression for \(y\): for each value of \(x\) (except \(x = 5\)), there is exactly one corresponding value of \(y\).
5. 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.

Key Concepts

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.