Question

which represents the reflection of $f(x) = \sqrt{x}$ over the x-axis?

x	$f(x)$
0	0
1	-1
4	-2

x	$f(x)$
0	0
1	undefined
4	undefined

Explanation:

Step1: Recall x-axis reflection rule

To reflect a function $f(x)$ over the x-axis, the transformed function is $g(x) = -f(x)$.

Step2: Define original function

Original function: $f(x) = \sqrt{x}$. Domain of $f(x)$ is $x \geq 0$ (since square root of negative numbers is undefined in real numbers).

Step3: Calculate transformed function

Transformed function: $g(x) = -\sqrt{x}$.

Step4: Verify values

• For $x=-1$: $g(-1) = -\sqrt{-1}$ (undefined)
• For $x=0$: $g(0) = -\sqrt{0} = 0$
• For $x=1$: $g(1) = -\sqrt{1} = -1$
• For $x=4$: $g(4) = -\sqrt{4} = -2$

Answer:

The first table (with values:
x=-1: undefined, x=0:0, x=1:-1, x=4:-2) represents the reflection of $f(x)=\sqrt{x}$ over the x-axis.