Question

match each graph below with the appropriate function.
a)
\\(\circ\\) \\(f(x) = \sqrt{x + 2}\\)
\\(\circ\\) \\(f(x) = -\sqrt{x + 2} - 2\\)
\\(\circ\\) \\(f(x) = -\sqrt{x + 2} + 2\\)
\\(\circ\\) \\(f(x) = \sqrt{x + 2} - 2\\)

Explanation:

Step1: Find domain of the graph

The graph starts at $x=-2$, so the domain is $x\geq -2$, which matches all options (since $\sqrt{x+2}$ requires $x+2\geq0 \implies x\geq-2$).

Step2: Identify starting point

The graph's starting point is $(-2, -2)$. Test $x=-2$ in each function:

• $f(-2)=\sqrt{-2+2}=0$ → $( -2, 0)$ (does not match)
• $f(-2)=-\sqrt{-2+2}-2=0-2=-2$ → $(-2, -2)$ (matches)
• $f(-2)=-\sqrt{-2+2}+2=0+2=2$ → $(-2, 2)$ (does not match)
• $f(-2)=\sqrt{-2+2}-2=0-2=-2$ → $(-2, -2)$ (matches)

Step3: Test another point

Take $x=2$. The graph at $x=2$ is around $-4$.

• For $f(x)=-\sqrt{x+2}-2$: $f(2)=-\sqrt{2+2}-2=-2-2=-4$ (matches)
• For $f(x)=\sqrt{x+2}-2$: $f(2)=\sqrt{2+2}-2=2-2=0$ (does not match)

Step4: Check direction

The graph decreases as $x$ increases, which matches the negative leading coefficient of $-\sqrt{x+2}-2$.

Answer:

$\boldsymbol{f(x) = -\sqrt{x+2} - 2}$