In Exercises 11–26, determine whether each equation defines y as a function of x. x = y²

Verified step by step guidance

1. Recall the definition of a function: for each input value of \( x \), there must be exactly one output value of \( y \).
2. Given the equation \( y^2 = x \), solve for \( y \) in terms of \( x \) by taking the square root of both sides: \( y = \pm \sqrt{x} \).
3. Notice that for each positive value of \( x \), there are two possible values of \( y \) (one positive and one negative), which means \( y \) is not uniquely determined by \( x \).
4. For \( x = 0 \), there is exactly one value of \( y \) (which is 0), but for other values of \( x \geq 0 \), there are two values of \( y \).
5. Since the equation does not assign exactly one \( y \) for each \( x \), it does not define \( y \) as a function of \( x \).

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. If any x maps to multiple y-values, the relation is not a function.

Solving Equations for y

To analyze if y is a function of x, we often solve the equation for y explicitly. For y² = x, solving for y gives y = ±√x, indicating two possible y-values for each positive x. This suggests the relation may not define y as a function of x.

Vertical Line Test

The vertical line test is a graphical method to determine if a relation is a function. If any vertical line intersects the graph more than once, the relation is not a function. For y² = x, the graph is a sideways parabola, failing the vertical line test.