The given function is:

$f(x) = |x|$

and the domain is restricted to:

$-\frac{\pi}{2} < x < \frac{\pi}{2}$

Key Features of $f(x) = |x|$:

1. Definition of the absolute value function:

   • For $x \geq 0$, $f(x) = x$.
   • For $x < 0$, $f(x) = -x$.

2. Domain: $-\frac{\pi}{2} < x < \frac{\pi}{2}$. This is an open interval, so the function is defined for values of $x$ strictly between $-\frac{\pi}{2}$ and $\frac{\pi}{2}$.

3. Range: Since the absolute value function outputs non-negative values, the range of $f(x)$ for the given domain is $0 \leq f(x) < \frac{\pi}{2}$.

4. Continuity: The function is continuous across the given interval, though it has a sharp corner at $x = 0$ (a non-differentiable point).

5. Symmetry: The function is symmetric with respect to the y-axis because $f(x) = f(-x)$ for all $x$ in the domain.

6. Derivative:

   • For $x > 0$, $f'(x) = 1$.
   • For $x < 0$, $f'(x) = -1$.
   • At $x = 0$, the derivative does not exist due to the sharp corner (non-differentiability).

Would you like to explore any specific aspect of this function or ask for a graph or further details?

Here are 5 related questions for deeper understanding:

1. What is the derivative of the absolute value function at $x = 0$?
2. How would the function change if the domain were extended to $-\infty < x < \infty$?
3. What is the graphical interpretation of the sharp corner at $x = 0$?
4. How can we express $f(x) = |x|$ as a piecewise function?
5. Can you describe the effect of shifting the function horizontally or vertically?

Tip: Absolute value functions often create V-shaped graphs and can be nondifferentiable at points where the argument equals zero.