For the piecewise defined function given in the example, in which definition would we plug in -1 to evaluate \( f(-1) \) ? the bottom part of the definition: \( x^{2} \) the function is not defined for \( f(-1) \) the middle part of the definition: 2 the top part of the definition: \( -x+1 \)

To determine which part of the piecewise function to use for evaluating \( f(-1) \), we need to understand the conditions under which each part of the function is applied. While the exact piecewise definition from the example isn't provided, we can infer based on the options given. Typically, a piecewise function is defined with different expressions based on the value of \( x \). The options you've provided suggest that the function might be divided into intervals around \( x = -1 \). Here's a plausible structure based on your options: \[ f(x) = \begin{cases} -x + 1 & \text{if } x < -1 \quad \text{(top part)} \\ 2 & \text{if } -1 \leq x \leq 1 \quad \text{(middle part)} \\ x^2 & \text{if } x > 1 \quad \text{(bottom part)} \end{cases} \] Given this structure: - **Top part (\(-x + 1\))** is used when \( x < -1 \). - **Middle part (2)** is used when \( -1 \leq x \leq 1 \). - **Bottom part (\(x^2\))** is used when \( x > 1 \). Since \( -1 \) falls within the interval \( -1 \leq x \leq 1 \), you would use the **middle part** of the definition to evaluate \( f(-1) \). \[ f(-1) = 2 \] **Answer:** Use the middle part of the definition, where f ( −1 ) is equal to 2.