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

To analyze this function, let's go over a few important details:

1. Domain: The square root function $\sqrt{x}$ is defined only for $x \geq 0$ (non-negative values), so the domain of $f(x) = -\sqrt{x}$ is also $x \geq 0$.

2. Range: Since the function takes the negative of the square root, the range will be all non-positive real numbers, $f(x) \leq 0$, or $(-\infty, 0]$.

3. Behavior:

   • At $x = 0$, $f(0) = -\sqrt{0} = 0$.
   • As $x$ increases, $\sqrt{x}$ increases, but since it is negated, $f(x)$ decreases.
   • The function is decreasing because as $x$ gets larger, $f(x)$ becomes more negative.

4. Graph Shape: The graph of $f(x) = -\sqrt{x}$ will look like a downward-opening curve starting at the origin (0,0) and moving leftward as $x$ increases.

Would you like a detailed graph or further explanation of any part of this function?

Here are some follow-up questions for deeper exploration:

1. What is the derivative of $f(x) = -\sqrt{x}$ and how does it describe the function's slope?
2. How would the function behave if we used $+\sqrt{x}$ instead of $-\sqrt{x}$?
3. What would the range be if the function was $f(x) = -\sqrt{x+1}$ instead?
4. Can we determine the concavity of $f(x) = -\sqrt{x}$ from its second derivative?
5. How would the graph of $f(x) = -\sqrt{x} + c$ change as $c$ varies?

Tip: To explore the graph behavior, start with specific values of $x$ and plot corresponding $f(x)$ values. This helps visualize the function's trend.