Ch. 2 - Functions and Graphs

Blitzer8th EditionCollege AlgebraISBN: 9780136970514Not the one you use?Change textbook

Blitzer 8th Edition

Ch. 2 - Functions and Graphs

Problem 20

Chapter 3, Problem 20

Determine whether each equation defines y as a function of x. $y = - \sqrt{x + 4}$

Verified step by step guidance

Recall that a relation defines y as a function of x if for every x-value in the domain, there is exactly one corresponding y-value.

Look at the given equation: $y = - \sqrt{x} + 4$. The square root function $\sqrt{x}$ is defined only for $x \geq 0$.

Since the square root function outputs only the principal (non-negative) root, $\sqrt{x}$ is always non-negative, so $-\sqrt{x}$ is always non-positive.

For each $x \geq 0$, the expression $-\sqrt{x} + 4$ produces exactly one value of $y$, because the square root function is single-valued and the operations are well-defined.

Therefore, the equation defines $y$ as a function of $x$ because each $x$ in the domain corresponds to exactly one $y$.

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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). To determine if an equation defines y as a function of x, check if for every x there is only one y. If multiple y-values exist for a single x, it is not a function.

Square Root Function and Domain

The square root function, √x, is defined only for x ≥ 0 because the square root of a negative number is not real. When analyzing y = -√x + 4, the domain is restricted to x ≥ 0, ensuring the expression under the root is valid.

Evaluating Function Output Uniqueness

To verify if y = -√x + 4 defines y as a function of x, substitute values of x and observe the output y. Since the square root yields a single non-negative value and the equation applies a negative sign and addition, each x produces exactly one y, confirming it is a function.

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