Determine whether each relation defines y as a function of x. Give the domain and range. See Example 5. y=√(4x+1)

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 \( x \) in the domain, there is exactly one \( y \).
2. Step 2: Analyze the given equation \( y = \sqrt{4x + 1} \). The square root function is defined only for non-negative values, so \( 4x + 1 \geq 0 \).
3. Step 3: Solve the inequality \( 4x + 1 \geq 0 \) to find the domain of the function. Subtract 1 from both sides to get \( 4x \geq -1 \), then divide by 4 to find \( x \geq -\frac{1}{4} \).
4. Step 4: Determine the range of the function. Since the square root function outputs non-negative values, the range of \( y = \sqrt{4x + 1} \) is \( y \geq 0 \).
5. Step 5: Conclude that \( y = \sqrt{4x + 1} \) defines \( y \) as a function of \( x \) because for each \( x \) in the domain \( x \geq -\frac{1}{4} \), there is exactly one corresponding \( y \). The domain is \( x \geq -\frac{1}{4} \) and the range is \( y \geq 0 \).

Key Concepts

Function Definition

A function is a relation where each input (x-value) corresponds to exactly one output (y-value). 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. For example, the relation y = √(4x + 1) is a function because for each x, there is only one corresponding y.

Domain

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. For the function y = √(4x + 1), the domain is determined by the condition 4x + 1 ≥ 0, leading to x ≥ -1/4.

Range

The range of a function is the set of all possible output values (y-values) that result from the domain. For the function y = √(4x + 1), since the square root function only produces non-negative outputs, the range is y ≥ 0. This means that as x varies within the domain, y will always be zero or greater.