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. For a specific \( x \) value greater than or equal to 2 (since \( x - 2 \) must be non-negative for the square root to be real), there are two corresponding \( y \) values: \( +\sqrt{x - 2} \) and \( -\sqrt{x - 2} \).
4. Because there are two outputs for some inputs, this violates the definition of a function, which requires exactly one output per input.
5. Therefore, the equation \( y = \pm \sqrt{x - 2} \) does not define \( y \) as a function of \( x \).

Key Concepts

Definition of a Function

A function assigns exactly one output value for each input value. In other words, for every x in the domain, there must be only one corresponding y. If an equation produces multiple y-values for a single x, 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 (where x ≥ 2), which can violate the definition of a function since one input corresponds to two outputs.

Domain Restrictions

The domain of the equation y = ±√(x-2) is limited to x ≥ 2 because the expression under the square root must be non-negative. Understanding domain restrictions helps determine where the function or relation is defined and whether it behaves as a function.