Determine whether each relation defines y as a function of x. Give the domain and range. See Example 5. 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 should be exactly one corresponding \(y\) value. Since \(y = -\sqrt{x}\) gives one unique output for each \(x \geq 0\), it is a function.
4. Find the range by considering the possible values of \(y\). Since \(y\) is the negative square root, \(y\) will be less than or equal to zero. So, 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 = -√x defines y as a function of x, check if for every x in the domain there is only one y value. Since the square root function outputs a single non-negative value and the negative sign just changes its sign, this relation assigns one y for each x.

Domain of a Function

The domain is the set of all possible input values (x) for which the function is defined. For y = -√x, the expression under the square root must be non-negative, so x must be greater than or equal to zero. Thus, the domain is all real numbers x ≥ 0.

Range of a Function

The range is the set of all possible output values (y) of the function. Since y = -√x, and √x is always non-negative, y will be the negative of a non-negative number, making y ≤ 0. Therefore, the range is all real numbers y ≤ 0.