Determine whether each equation defines y as a function of x. y = ±√(x-2)

Verified step by step guidance

1. Understand the definition of a function: A relation is a function if each input (x-value) corresponds to exactly one output (y-value).
2. Examine the given equation: \( y = \pm \sqrt{x-2} \).
3. Recognize that the \( \pm \) symbol indicates two possible values for \( y \) for each \( x \) where \( x \geq 2 \).
4. Consider the implications: For any \( x \geq 2 \), there are two possible values for \( y \) (one positive and one negative), which means \( y \) is not uniquely determined by \( x \).
5. Conclude that the equation does not define \( y \) as a function of \( x \) because it violates the definition of a function by providing two outputs for a single input.

Key Concepts

Function Definition

A function is a relation that assigns exactly one output (y) for each input (x). For an equation to define y as a function of x, every x-value must correspond to a single y-value. If an x-value can produce multiple y-values, the relation is not a function.

Vertical Line Test

The vertical line test is a visual way to determine if a graph represents a function. If any vertical line intersects the graph at more than one point, the relation is not a function. This test helps to quickly assess whether y can be uniquely determined from x.

Square Root Function

The square root function, represented as y = ±√(x-2), indicates that for each x-value greater than or equal to 2, there are two corresponding y-values: one positive and one negative. This duality means that the equation does not define y as a function of x, as it fails the definition of a function.