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

Verified step by step guidance

1. Recall the definition of a function: for each input value \( x \), there must be exactly one output value \( y \).
2. Examine the given equation: \( y = \pm \sqrt{x - 2} \). The \( \pm \) symbol means there are two possible values of \( y \) for each \( x \) (one positive and one negative).
3. Consider a specific value of \( x \) where \( x - 2 \geq 0 \) (to keep the expression under the square root defined). For such \( x \), there are two corresponding \( y \) values: \( +\sqrt{x - 2} \) and \( -\sqrt{x - 2} \).
4. Since there are two outputs for a single input \( x \), this violates the definition of a function.
5. Therefore, conclude that the equation \( y = \pm \sqrt{x - 2} \) 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). If an equation assigns more than one y-value to a single x-value, it does not define y as a function of x.

Square Root and ± Notation

The expression ±√(x-2) means both the positive and negative square roots are considered. This implies two possible y-values for each x ≥ 2, which affects whether the relation is a function.

Domain Restrictions

The domain of y = ±√(x-2) is x ≥ 2 because the expression under the square root must be non-negative. Understanding domain restrictions helps determine valid inputs and analyze the function behavior.