Ch. 2 - Graphs and Functions

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

Lial 13th Edition

Ch. 2 - Graphs and Functions

Problem 42

Chapter 3, Problem 42

Determine whether each relation defines y as a function of x. Give the domain and range. y=-√x

Verified step by step guidance

Identify the given relation: $y = -\sqrt{x}$. This means $y$ is defined as the negative square root of $x$.

Determine the domain by considering the values of $x$ for which the expression under the square root is defined. Since the square root function requires the radicand to be non-negative, set $x \geq 0$.

Check if the relation defines $y$ as a function of $x$. For each $x$ in the domain, there is exactly one value of $y$ because the square root function returns only the principal (non-negative) root, and the negative sign in front makes $y$ unique for each $x$.

Find the range by evaluating the possible values of $y$. Since $y = -\sqrt{x}$ and $\sqrt{x} \geq 0$, $y$ will be less than or equal to zero. Thus, the range is $y \leq 0$.

Summarize: The relation defines $y$ as a function of $x$ with domain $[0, \infty)$ and range $(-\infty, 0]$.

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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 corresponds to exactly one output y. To determine if y is a function of x, check that for every x-value there is only one y-value. If any x maps to multiple y-values, the relation is not a function.

Domain of a Function

The domain is the set of all possible input values (x-values) for which the function is defined. For y = -√x, the expression under the square root must be non-negative, so the domain includes all x ≥ 0.

Range of a Function

The range is the set of all possible output values (y-values) of the function. Since y = -√x produces non-positive values (because of the negative sign), the range is all y ≤ 0.

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