1. **State the problem:** We have a piecewise function defined as: $$f(x) = \begin{cases} 3x & \text{if } x < 0 \\ x + 1 & \text{if } 0 \leq x \leq 2 \\ (x - 2)^2 & \text{if } x > 2 \end{cases}$$ We need to find the values of $f(-5)$, $f(0)$, $f(1)$, $f(2)$, and $f(5)$. 2. **Recall the rules for piecewise functions:** - Identify which piece of the function applies based on the value of $x$. - Substitute $x$ into the correct expression. 3. **Calculate each value:** - For $f(-5)$: Since $-5 < 0$, use $f(x) = 3x$. $$f(-5) = 3 \times (-5) = -15$$ - For $f(0)$: Since $0 \leq 0 \leq 2$, use $f(x) = x + 1$. $$f(0) = 0 + 1 = 1$$ - For $f(1)$: Since $0 \leq 1 \leq 2$, use $f(x) = x + 1$. $$f(1) = 1 + 1 = 2$$ - For $f(2)$: Since $0 \leq 2 \leq 2$, use $f(x) = x + 1$. $$f(2) = 2 + 1 = 3$$ - For $f(5)$: Since $5 > 2$, use $f(x) = (x - 2)^2$. $$f(5) = (5 - 2)^2 = 3^2 = 9$$ 4. **Final answers:** $$f(-5) = -15, \quad f(0) = 1, \quad f(1) = 2, \quad f(2) = 3, \quad f(5) = 9$$ This completes the evaluation of the piecewise function at the given points.