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

Verified step by step guidance

1. Identify the given relation: \(y = -\sqrt{x}\). This means \(y\) is defined as the negative square root of \(x\).
2. 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 \(x \geq 0\).
3. 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 \(x\).
4. 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 \(y \leq 0\).
5. Summarize: The relation defines \(y\) as a function of \(x\) with domain \([0, \infty)\) and range \((-\infty, 0]\).

Key Concepts

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.