1. **Problem Statement:** Given the graph and points, find the values of $f(-2)$, $\lim_{x \to 0^-} f(x)$, $\lim_{x \to 0^+} f(x)$, determine if $\lim_{x \to 0} f(x)$ exists, and find $f(-6)$, $f(2)$, $g(5)$, and $f(-1)$. 2. **Understanding the graph:** - There is a black filled circle at $(-1,5)$, so $f(-1) = 5$. - An orange horizontal arrow points leftward from $(-1,5)$, indicating the function extends left from that point at $y=5$. - An orange diagonal arrow starts near $(-4,-4)$ and goes upward right to about $(6,3)$, suggesting a function $g(x)$ increasing from $-4$ to $3$ over that interval. 3. **Evaluate $f(-2)$:** Since the horizontal arrow extends left from $(-1,5)$, the function $f$ is constant at $y=5$ for $x < -1$. Thus, $f(-2) = 5$. 4. **Evaluate $\lim_{x \to 0^-} f(x)$:** For $x$ approaching $0$ from the left, the function $f$ is on the horizontal line $y=5$ (since $0 > -1$ but the graph suggests the function is constant left of $-1$ and no other info is given). So, $\lim_{x \to 0^-} f(x) = 5$. 5. **Evaluate $\lim_{x \to 0^+} f(x)$:** From the right side of $0$, the graph is not explicitly described for $f$, but the diagonal arrow likely represents $g(x)$, not $f(x)$. Without further info, assume $f$ is not defined or different on the right side of $0$. So, $\lim_{x \to 0^+} f(x)$ is not given or does not equal $5$. 6. **Does $\lim_{x \to 0} f(x)$ exist?** Since the left and right limits at $0$ are not equal ($5$ vs unknown or different), $\lim_{x \to 0} f(x)$ does not exist. 7. **Evaluate $f(-6)$:** Since $f$ is constant at $y=5$ for $x < -1$, $f(-6) = 5$. 8. **Evaluate $f(2)$:** No explicit info about $f$ at $x=2$ is given, but the diagonal arrow is for $g$, so $f(2)$ is undefined or not given. 9. **Evaluate $g(5)$:** The diagonal arrow from $(-4,-4)$ to $(6,3)$ suggests $g$ is a line passing through these points. The slope is $m = \frac{3 - (-4)}{6 - (-4)} = \frac{7}{10} = 0.7$. The line equation is $g(x) = m(x + 4) - 4 = 0.7(x + 4) - 4$. Calculate $g(5) = 0.7(5 + 4) - 4 = 0.7 \times 9 - 4 = 6.3 - 4 = 2.3$. 10. **Evaluate $f(-1)$:** Given by the black filled circle, $f(-1) = 5$. **Final answers:** $$f(-2) = 5$$ $$\lim_{x \to 0^-} f(x) = 5$$ $$\lim_{x \to 0^+} f(x) \text{ does not equal } 5 \text{ (not given)}$$ $$\lim_{x \to 0} f(x) \text{ does not exist}$$ $$f(-6) = 5$$ $$f(2) \text{ not given}$$ $$g(5) = 2.3$$ $$f(-1) = 5$$