1. **State the problem:** We have a piecewise function: $$f(x) = \begin{cases} 1 - x & \text{if } x \leq -1 \\ x^{2} & \text{if } x > -1 \end{cases}$$ We need to evaluate $f(-2)$, $f(-1)$, and $f(0)$, and then write the function represented by the described graph. 2. **Evaluate $f(-2)$:** Since $-2 \leq -1$, use the first case: $$f(-2) = 1 - (-2) = 1 + 2 = 3$$ 3. **Evaluate $f(-1)$:** Since $-1 \leq -1$, use the first case: $$f(-1) = 1 - (-1) = 1 + 1 = 2$$ 4. **Evaluate $f(0)$:** Since $0 > -1$, use the second case: $$f(0) = 0^{2} = 0$$ 5. **Write the function from the graph description:** - For $x$ from $-4$ to $-1$, the graph is a line segment from $(-4, 2)$ to $(-1, -1)$. The slope $m$ is: $$m = \frac{-1 - 2}{-1 - (-4)} = \frac{-3}{3} = -1$$ Equation of the line: $$y - 2 = -1(x + 4) \Rightarrow y = -x - 2$$ - For $x$ from $-1$ to $0$, the graph is a horizontal line at $y = -1$. - For $x > 0$, the graph is a parabola $y = x^{2}$. 6. **Combine the piecewise function $t(x)$ from the graph:** $$t(x) = \begin{cases} -x - 2 & -4 \leq x < -1 \\ -1 & -1 \leq x \leq 0 \\ x^{2} & x > 0 \end{cases}$$ **Final answers:** $$f(-2) = 3, \quad f(-1) = 2, \quad f(0) = 0$$ $$t(x) = \begin{cases} -x - 2 & -4 \leq x < -1 \\ -1 & -1 \leq x \leq 0 \\ x^{2} & x > 0 \end{cases}$$