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: Identify the type of equation given. The equation is y = \sqrt{4x + 1}, which is a square root function.
2. Step 2: Determine if y is a function of x. For y to be a function of x, each x-value must correspond to exactly one y-value. Since the square root function outputs a single non-negative value for each input, y is a function of x.
3. Step 3: Find the domain of the function. The expression inside the square root, 4x + 1, must be greater than or equal to zero for the square root to be defined. Solve the inequality 4x + 1 \geq 0 to find the domain.
4. Step 4: Solve the inequality 4x + 1 \geq 0. Subtract 1 from both sides to get 4x \geq -1, then divide by 4 to find x \geq -\frac{1}{4}. Thus, the domain is x \geq -\frac{1}{4}.
5. Step 5: 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.

Key Concepts

Function Definition

A relation defines y as a function of x if each input x corresponds to exactly one output y. This means that for every value of x in the domain, there is a unique value of y. To determine if a relation is a function, one can use the vertical line test, which states that if a vertical line intersects the graph of the relation at more than one point, it is not a function.

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 inconsistencies, 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, as the square root of a negative number is not a real number. In the given relation y = √(4x + 1), the expression inside the square root must be greater than or equal to zero to yield real values for y. This condition helps determine the domain of the function.