Let's compute the value of the function $f(x) = |x| - x^2 - \lfloor x \rfloor$ at $x = -0.5$.

Step-by-step solution:

1. Absolute value term: $|x| = |-0.5| = 0.5$

2. Square term: $x^2 = (-0.5)^2 = 0.25$

3. Floor function: The floor function $\lfloor x \rfloor$ is the greatest integer less than or equal to $x$. Since $x = -0.5$, the greatest integer less than or equal to $-0.5$ is $-1$. $\lfloor -0.5 \rfloor = -1$

4. Putting it all together: $f(-0.5) = | -0.5 | - (-0.5)^2 - \lfloor -0.5 \rfloor = 0.5 - 0.25 - (-1)$ $f(-0.5) = 0.5 - 0.25 + 1 = 1.25$

So, the value of $f(-0.5)$ is 1.25.

Do you have any questions or need further details?

Here are five related questions to consider:

1. How does the function $f(x)$ behave for positive values of $x$?
2. What is the derivative of $f(x)$ for values where $x \neq 0$?
3. How does the floor function $\lfloor x \rfloor$ impact $f(x)$ for fractional values of $x$?
4. What would the value of $f(x)$ be for $x = 1.5$?
5. Can you plot the graph of $f(x)$ for $x \in [-2, 2]$?

Tip: The floor function always rounds down to the nearest integer, regardless of whether the input is positive or negative.