The function \( t(x) = \frac{1}{2} |x| \) is a piecewise function that represents a transformation of the absolute value function \( |x| \). 

### Key Points:
1. **Absolute Value**: The absolute value function \( |x| \) returns the non-negative value of \( x \). 
   - For \( x \geq 0 \), \( |x| = x \).
   - For \( x < 0 \), \( |x| = -x \).

2. **Scaling by \(\frac{1}{2}\)**: The function scales the output of the absolute value function by a factor of \(\frac{1}{2}\). This means that the graph of the function \( t(x) = \frac{1}{2} |x| \) will be a V-shape, similar to the graph of \( |x| \), but it will be "compressed" vertically.

### Piecewise Representation:
\[ t(x) = 
\begin{cases} 
\frac{1}{2}x & \text{if } x \geq 0 \\
\frac{1}{2}(-x) & \text{if } x < 0 
\end{cases} \]

### Graph:
- For \( x \geq 0 \), the function is a line with slope \( \frac{1}{2} \) passing through the origin.
- For \( x < 0 \), the function is a line with slope \( -\frac{1}{2} \) passing through the origin.

### Properties:
- **Domain**: The domain of \( t(x) \) is all real numbers, \( \mathbb{R} \).
- **Range**: The range of \( t(x) \) is \( [0, \infty) \), since the absolute value ensures non-negative outputs, and the scaling does not change this.
- **Continuity**: The function is continuous for all \( x \), with no breaks or jumps.

### Graphical Interpretation:
The graph is a V-shaped curve centered at the origin, but with the slopes \( \pm \frac{1}{2} \) instead of \( \pm 1 \), making it less steep than the standard \( |x| \) graph.

Would you like more details or have any questions?

Here are some related questions:
1. What are some real-world applications of absolute value functions?
2. How does changing the coefficient of \( |x| \) affect the graph?
3. What would the function look like if we added a constant to \( t(x) \)?
4. How do you find the derivative of \( t(x) = \frac{1}{2}|x| \)?
5. Can you solve \( t(x) = 2 \) for \( x \)?
6. What happens if we apply a horizontal shift to \( t(x) \)?
7. How does this function behave as \( x \) approaches infinity or negative infinity?
8. Can you find the integral of \( t(x) \) with respect to \( x \)?

**Tip**: Always consider the impact of transformations like scaling, shifting, and reflecting on the basic graph of a function to understand how the graph changes.

Explore the piecewise function t(x) = 1/2 |x|, which transforms the absolute value function by scaling its outputs. Learn about its piecewise representation, graph characteristics, domain, range, and continuity. Suitable for high school students studying functions and transformations.

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