If `f(x)={{:(2x-1,"when "xle0),(x^(2),"when "xgt0):},"then find "f((1)/(2)) and f((-1)/(2))`

Answer

To solve the problem, we need to evaluate the function \( f(x) \) at two different points: \( f\left(\frac{1}{2}\right) \) and \( f\left(-\frac{1}{2}\right) \). The function \( f(x) \) is defined piecewise as follows:

\[ f(x) = \begin{cases} 2x - 1 & \text{when } x \leq 0 \\ x^2 & \text{when } x > 0 \end{cases} \]

### Step 1: Calculate \( f\left(\frac{1}{2}\right) \)

1. **Identify the condition**: Since \( \frac{1}{2} > 0 \), we will use the second part of the piecewise function, which is \( f(x) = x^2 \).
2. **Substitute \( x = \frac{1}{2} \)**:
\[ f\left(\frac{1}{2}\right) = \left(\frac{1}{2}\right)^2 = \frac{1}{4} \]

### Step 2: Calculate \( f\left(-\frac{1}{2}\right) \)

1. **Identify the condition**: Since \( -\frac{1}{2} \leq 0 \), we will use the first part of the piecewise function, which is \( f(x) = 2x - 1 \).
2. **Substitute \( x = -\frac{1}{2} \)**:
\[ f\left(-\frac{1}{2}\right) = 2\left(-\frac{1}{2}\right) - 1 = -1 - 1 = -2 \]

### Final Answers

- \( f\left(\frac{1}{2}\right) = \frac{1}{4} \)
- \( f\left(-\frac{1}{2}\right) = -2 \)