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. Step 1: Understand the definition of a function. A relation defines y as a function of x if for every x-value, there is exactly one corresponding y-value.
2. Step 2: Analyze the given relation y = -\sqrt{x}. Notice that for each x-value, there is only one possible y-value, which is -\sqrt{x}. Therefore, this relation does define y as a function of x.
3. Step 3: Determine the domain of the function. Since we have a square root, x must be greater than or equal to 0 to avoid taking the square root of a negative number. Thus, the domain is x \geq 0.
4. Step 4: Determine the range of the function. Since y = -\sqrt{x}, the output y will always be non-positive (less than or equal to 0). Therefore, the range is y \leq 0.
5. Step 5: Summarize the findings: The relation y = -\sqrt{x} defines y as a function of x with a domain of x \geq 0 and a range of y \leq 0.

Key Concepts

Function Definition

A function is a relation where each input (x-value) corresponds to exactly one output (y-value). This means that for any given x, there cannot be multiple y-values. To determine if a relation defines y as a function of x, we check if any x-value is paired with more than one y-value.

Domain and Range

The domain of a function is the set of all possible input values (x-values) that can be used without causing any mathematical issues, such as division by zero or taking the square root of a negative number. The range is the set of all possible output values (y-values) that result from the function. Understanding the domain and range is crucial for analyzing the behavior of the function.

Square Root Function

The square root function, represented as y = -√x, is defined only for non-negative values of x, since the square root of a negative number is not a real number. This function produces non-positive outputs because of the negative sign in front of the square root. Analyzing this function helps in determining its domain (x ≥ 0) and range (y ≤ 0).