\( f(x)=\left\{\begin{array}{ll}|x| \text { if } x \leq 0 & \text { a. Find } f(-2), f(-5), \text { and } f(2) \text {. } \\ \frac{1}{x} \text { if } x>0 & \text { b. Sketch the graph of the piecewise-de } \\ \text { c. Determine the domain of } f .\end{array}\right. \)

To analyze the piecewise function \( f(x) \), we will address each part of the problem step by step. ### a. Find \( f(-2) \), \( f(-5) \), and \( f(2) \). 1. **Finding \( f(-2) \)**: Since \(-2 \leq 0\), we use the first piece of the function: \[ f(-2) = |-2| = 2. \] 2. **Finding \( f(-5) \)**: Since \(-5 \leq 0\), we again use the first piece of the function: \[ f(-5) = |-5| = 5. \] 3. **Finding \( f(2) \)**: Since \(2 > 0\), we use the second piece of the function: \[ f(2) = \frac{1}{2} = 0.5. \] ### Summary of results: - \( f(-2) = 2 \) - \( f(-5) = 5 \) - \( f(2) = 0.5 \) ### b. Sketch the graph of the piecewise-defined function. To sketch the graph of \( f(x) \): 1. For \( x \leq 0 \), the function is \( f(x) = |x| \). This is a V-shaped graph that opens upwards, starting from the origin (0,0) and extending to the left. The points on this part of the graph include: - \( f(-2) = 2 \) (point (-2, 2)) - \( f(-5) = 5 \) (point (-5, 5)) 2. For \( x > 0 \), the function is \( f(x) = \frac{1}{x} \). This is a hyperbola that approaches the x-axis as \( x \) increases. The point on this part of the graph includes: - \( f(2) = 0.5 \) (point (2, 0.5)) ### Graph: - The left side (for \( x \leq 0 \)) will be a straight line from the origin to the left, with points at (-2, 2) and (-5, 5). - The right side (for \( x > 0 \)) will be a curve starting from (0, ∞) and approaching the x-axis as \( x \) increases, passing through (2, 0.5). ### c. Determine the domain of \( f \). The domain of \( f \) is the set of all \( x \) values for which \( f(x) \) is defined. - For \( x \leq 0 \), \( f(x) = |x| \) is defined for all \( x \leq 0 \). - For \( x > 0 \), \( f(x) = \frac{1}{x} \) is defined for all \( x > 0\). Thus, the domain of \( f \) is: \[ \text{Domain of } f = (-\infty, 0] \cup (0, \infty). \] ### Final Summary: - \( f(-2) = 2 \) - \( f(-5) = 5 \) - \( f(2) = 0.5 \) - Domain of \( f \): \( (-\infty, 0] \cup (0, \infty) \)