In Exercises 11–26, determine whether each equation defines y as a function of x. x + y = 16

Verified step by step guidance

1. Start by understanding the problem: we need to determine if the equation \(x + y = 16\) defines \(y\) as a function of \(x\). A function means for each \(x\) there is exactly one \(y\) value.
2. Rewrite the equation to solve for \(y\) in terms of \(x\). Subtract \(x\) from both sides to isolate \(y\): \(y = 16 - x\).
3. Observe the expression \(y = 16 - x\). For every value of \(x\), there is exactly one corresponding value of \(y\) calculated by subtracting \(x\) from 16.
4. Since each \(x\) corresponds to exactly one \(y\), the equation defines \(y\) as a function of \(x\).
5. Conclude that the equation \(x + y = 16\) does define \(y\) as a function of \(x\) because it can be expressed as \(y = 16 - x\), which passes the vertical line test.

Key Concepts

Definition of a Function

A function is a relation where each input (x-value) corresponds to exactly one output (y-value). This means for every x, there is only one y. Understanding this helps determine if an equation defines y as a function of x.

Solving for y in Terms of x

To check if y is a function of x, solve the equation for y explicitly. If y can be expressed as a single formula involving x without ambiguity, then y is a function of x.

Vertical Line Test

The vertical line test is a graphical method to determine if y is a function of x. If any vertical line intersects the graph of the equation more than once, y is not a function of x.