Question

b)

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

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

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

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

Explanation:

Step1: Identify the parent function

The parent function is $y=\sqrt{x}$, which has a domain $x\geq0$ and starts at $(0,0)$, increasing rightward.

Step2: Analyze horizontal shift

The graph starts at $x=-2$, so there is a left shift by 2 units. This transforms the function to $y=\sqrt{x+2}$.

Step3: Analyze vertical shift and direction

The starting point of the graph is $(-2, 2)$? No, correct: the graph at $x=-2$ has $y=2$? Wait no, check the graph: at $x=-2$, $y=2$; at $x=2$, $y=0$. Test $f(x)=-\sqrt{x+2}+2$:
For $x=-2$: $f(-2)=-\sqrt{-2+2}+2=0+2=2$, matches.
For $x=2$: $f(2)=-\sqrt{2+2}+2=-2+2=0$, matches.
For $x=7$: $f(7)=-\sqrt{7+2}+2=-3+2=-1$, matches the downward trend.
Eliminate other options:

• $f(x)=\sqrt{x+2}$: at $x=-2$, $y=0$, does not match.
• $f(x)=-\sqrt{x+2}-2$: at $x=-2$, $y=-2$, does not match.
• $f(x)=\sqrt{x-2}-2$: domain $x\geq2$, does not match the graph starting at $x=-2$.

Answer:

$\boldsymbol{f(x) = -\sqrt{x+2} + 2}$ (Option 4)