Question

the function $f(x)= -sqrt{-x}$ is shown on the graph.
which statement is correct?
○ the range of the graph is all real numbers greater than or equal to 0.
○ the domain of the graph is all real numbers greater than or equal to 0.
○ the range and domain of the graph are the same.
○ the domain of the graph is all real numbers.

Explanation:

Step1: Find the domain

For $f(x)=-\sqrt{-x}$, the expression under the square root must be non-negative:
$-x \geq 0 \implies x \leq 0$.
So the domain is all real numbers $\leq 0$.

Step2: Find the range

Let $y = -\sqrt{-x}$. Since $\sqrt{-x} \geq 0$, multiplying by $-1$ reverses the inequality:
$y = -\sqrt{-x} \leq 0$.
So the range is all real numbers $\leq 0$.

Step3: Compare domain and range

Both the domain and range are all real numbers less than or equal to 0, so they are the same.

Answer:

The range and domain of the graph are the same.