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

Verified step by step guidance

1. Identify the given relation: \(y = \sqrt{4x + 1}\). This means \(y\) is defined as the square root of the expression \$4x + 1$.
2. Determine the domain by finding all values of \(x\) for which the expression inside the square root is non-negative, since the square root of a negative number is not a real number. Set up the inequality: \$4x + 1 \geq 0$.
3. Solve the inequality \$4x + 1 \geq 0\( to find the domain. Subtract 1 from both sides: \)4x \geq -1\(, then divide both sides by 4: \)x \geq -\frac{1}{4}$.
4. Check if the relation defines \(y\) as a function of \(x\). Since for each \(x\) in the domain there is exactly one non-negative value of \(y\) (the principal square root), this relation does define \(y\) as a function of \(x\).
5. Determine the range by considering the possible values of \(y\). Since \(y = \sqrt{4x + 1}\) and the square root function outputs only non-negative values, the smallest value of \(y\) is 0 (when \(x = -\frac{1}{4}\)), and \(y\) increases without bound as \(x\) increases. So, the range is \(y \geq 0\).

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. This ensures the relation passes the vertical line test when graphed.

Domain of a Function

The domain is the set of all possible input values (x-values) for which the function is defined. For functions involving square roots, the expression inside the root must be non-negative to yield real outputs, restricting the domain accordingly.

Range of a Function

The range is the set of all possible output values (y-values) of the function. For y = √(4x + 1), since square roots produce non-negative results, the range includes all y-values greater than or equal to zero.