Ch. 2 - Graphs and Functions

Lial13th EditionCollege AlgebraISBN: 9780136881063Not the one you use?Change textbook

Lial 13th Edition

Ch. 2 - Graphs and Functions

Problem 27

Chapter 3, Problem 27

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

Verified step by step guidance

Recall the definition of a function: for each input value $ x $, there must be exactly one output value $ y $.

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).

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} $.

Since there are two outputs for a single input $ x $, this violates the definition of a function.

Therefore, conclude that the equation $ y = \pm \sqrt{x - 2} $ does not define $ y $ as a function of $ x $.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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.

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