The function given in the image is $f(x) = -\sqrt{-x}$.

This function involves both the square root and a negative sign, so let's break it down:

1. The square root $\sqrt{-x}$ is only defined for $x \leq 0$ because the expression under the square root must be non-negative.
2. The negative sign in front of the square root reflects the graph downward. Normally, $\sqrt{-x}$ produces a positive value, but the negative sign makes it negative, so the graph will be flipped below the x-axis.

Domain:

Since $\sqrt{-x}$ is defined only when $-x \geq 0$, we know that $x \leq 0$. Hence, the domain is $(-\infty, 0]$.

Range:

For $x = 0$, $f(x) = 0$. As $x$ decreases (becomes more negative), the square root grows larger, but the negative sign in front ensures that $f(x)$ remains negative. Therefore, the range is $(-\infty, 0]$.

Table of values:

To create the graph, let's calculate a few points:

• For $x = 0$, $f(0) = -\sqrt{0} = 0$.
• For $x = -1$, $f(-1) = -\sqrt{1} = -1$.
• For $x = -4$, $f(-4) = -\sqrt{4} = -2$.
• For $x = -9$, $f(-9) = -\sqrt{9} = -3$.

Plotting the points:

• (0, 0)
• (-1, -1)
• (-4, -2)
• (-9, -3)

You can plot these points and connect them with a smooth curve, keeping in mind that the curve exists only in the left half of the graph (because $x \leq 0$) and curves downward.

Would you like more details or have any questions about this function or its graph?

Additional Questions:

1. What is the significance of the negative sign in front of the square root?
2. How does restricting the domain of a function like $f(x) = \sqrt{-x}$ affect its graph?
3. How can transformations (e.g., vertical reflection) be used to predict the graph's shape?
4. How do you determine the domain of functions involving square roots?
5. What are the common characteristics of graphs involving square root functions?

Tip:

Always check the domain of functions with square roots, as they require non-negative values under the radical to be defined.