Let `f(x)={{:(,3^x,at,-1lt x lt0),(,4,at,0 le x lt 1),(,3x-1, at,1 le x le 3):}`
Find f(2), f(0), f(0.5), f(-0.5), f(3).

Answer

To find the values of the function \( f(x) \) at specific points, we will evaluate \( f(2) \), \( f(0) \), \( f(0.5) \), \( f(-0.5) \), and \( f(3) \) based on the piecewise definition of the function:

\[ f(x) = \begin{cases} 3^x & \text{for } -1 < x < 0 \\ 4 & \text{for } 0 \leq x < 1 \\ 3x - 1 & \text{for } 1 \leq x \leq 3 \end{cases} \]

### Step-by-Step Solution:

1. **Finding \( f(2) \)**:
- Since \( 2 \) is in the range \( 1 \leq x \leq 3 \), we use the third case of the function:
\[ f(2) = 3(2) - 1 = 6 - 1 = 5 \]

2. **Finding \( f(0) \)**:
- Since \( 0 \) is in the range \( 0 \leq x < 1 \), we use the second case of the function:
\[ f(0) = 4 \]

3. **Finding \( f(0.5) \)**:
- Since \( 0.5 \) is in the range \( 0 < x < 1 \), we again use the second case of the function:
\[ f(0.5) = 4 \]

4. **Finding \( f(-0.5) \)**:
- Since \( -0.5 \) is in the range \( -1 < x < 0 \), we use the first case of the function:
\[ f(-0.5) = 3^{-0.5} = \frac{1}{\sqrt{3}} \approx 0.577 \]

5. **Finding \( f(3) \)**:
- Since \( 3 \) is in the range \( 1 \leq x \leq 3 \), we use the third case of the function:
\[ f(3) = 3(3) - 1 = 9 - 1 = 8 \]

### Summary of Results:
- \( f(2) = 5 \)
- \( f(0) = 4 \)
- \( f(0.5) = 4 \)
- \( f(-0.5) = \frac{1}{\sqrt{3}} \approx 0.577 \)
- \( f(3) = 8 \)