$$ f\left(x\right)=\left\vert x\right\vert $$, $$ -\frac{\pi}{2} $$$$ <x<\frac{\pi}{2} $$

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.

Explore the properties of the absolute value function f(x) = |x| defined on the domain (-π/2, π/2). Learn about its continuity, symmetry, range, and non-differentiability at x = 0. Suitable for high school students in Grades 10-12.

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