1. **State the problem:** We need to identify which piecewise function matches the given graph. 2. **Analyze each piece of the graph:** - Left piece: horizontal line at $y=5$ ending with a filled point at $(-2,5)$. - Middle piece: parabola with open circles at $(-2,-1)$ and $(1,-4)$. - Right piece: increasing curve starting with a filled point at $(1,-3)$ and rising steeply for $x>1$. 3. **Check each function's pieces:** **Option A:** - Left: $f(x)=5$ for $x \leq -2$ matches the horizontal line at $y=5$. - Middle: $f(x)=x^2+5$ for $-2 < x < 1$. Evaluate at $x=-2$: $(-2)^2+5=4+5=9$ (should be open circle at $(-2,9)$ but graph shows $(-2,-1)$ open circle, so no match). **Option B:** - Left: $f(x)=5$ for $x \leq -2$ matches horizontal line at $y=5$. - Middle: $f(x)=x^2-5$ for $-2 < x < 1$. Evaluate at $x=-2$: $(-2)^2-5=4-5=-1$ matches open circle at $(-2,-1)$. Evaluate at $x=1$: $1^2-5=1-5=-4$ matches open circle at $(1,-4)$. - Right: $f(x)=2(x-2)-2=2x-4-2=2x-6$ for $x \geq 1$. Evaluate at $x=1$: $2(1)-6=2-6=-4$ but graph shows filled point at $(1,-3)$, so no match. **Option C:** - Left: $f(x)=-5$ for $x \leq -2$ does not match horizontal line at $y=5$. 4. **Re-examine Option B's right piece:** The graph shows filled point at $(1,-3)$ but function gives $-4$. 5. **Check if the right piece in Option A matches:** - $f(x)=2(x+2)-2=2x+4-2=2x+2$ for $x \geq 1$. - Evaluate at $x=1$: $2(1)+2=4$ but graph shows $-3$, no match. 6. **Check Option C's right piece:** - $f(x)=2(x+2)-3=2x+4-3=2x+1$ for $x \geq 1$. - Evaluate at $x=1$: $2(1)+1=3$ but graph shows $-3$, no match. 7. **Conclusion:** The middle piece parabola matches Option B perfectly. The left piece matches Option B. The right piece in Option B evaluated at $x=1$ is $-4$ but graph shows $-3$. Since the graph's right piece starts at $(1,-3)$ filled point, none of the options perfectly match the right piece. **However, the graph's middle piece parabola is $y=x^2-5$ which matches Option B exactly, and the left piece matches Option B.** Therefore, the best match is **Option B**. **Final answer:** $$\boxed{\text{Option B}}$$