Solving x²-1/4: Finding Negative Function Values and Domains

To solve this problem, we will follow these steps:

• Step 1: Solve the equation $x^2 = \frac{1}{4}$ to find critical points.
• Step 2: Determine the sign of $f(x) = x^2 - \frac{1}{4}$ in each interval between and beyond these points.
• Step 3: Identify the interval where $f(x) < 0$.

Now, let's work through each step:
Step 1: Solve the equation $x^2 = \frac{1}{4}$. The solutions are $x = \frac{1}{2}$ and $x = -\frac{1}{2}$ because solving this gives $x = \pm \sqrt{\frac{1}{4}}$.
Step 2: To find where $f(x) < 0$, we need $x^2 < \frac{1}{4}$. This is equivalent to finding where $-\frac{1}{2} < x < \frac{1}{2}$. In this interval, $f(x)$ is negative.
Step 3: The function $y = x^2 - \frac{1}{4}$ is negative for values of $x$ between these roots, i.e., for $-\frac{1}{2} < x < \frac{1}{2}$.

Therefore, the solution is that the function is negative for $-\frac{1}{2} < x < \frac{1}{2}$.