The problem presents the equation $y = \sqrt{4x + 1}$. We need to analyze this equation and provide some analysis, which I assume is finding the domain and range.

1. Problem Description

The problem presents the equation $y = \sqrt{4x + 1}$. We need to analyze this equation and provide some analysis, which I assume is finding the domain and range.

2. Solution Steps

First, we need to determine the domain of the function. Since we have a square root, the expression inside the square root must be non-negative.

$4x + 1 \ge 0$

Subtract 1 from both sides:

$4x \ge -1$

Divide both sides by 4:

$x \ge -\frac{1}{4}$

So, the domain is $x \ge -\frac{1}{4}$.

Next, we need to determine the range of the function. Since the square root function always returns non-negative values, and $x \ge -\frac{1}{4}$, we have:

$y = \sqrt{4x + 1} \ge 0$

When $x = -\frac{1}{4}$, we have $y = \sqrt{4(-\frac{1}{4}) + 1} = \sqrt{-1 + 1} = \sqrt{0} = 0$.

As $x$ increases, $4x + 1$ also increases, and therefore, $\sqrt{4x + 1}$ increases. Thus, $y$ can take on any non-negative value.

So, the range is $y \ge 0$.

3. Final Answer

Domain: $x \ge -\frac{1}{4}$

Range: $y \ge 0$