Let's start by interpreting the given function. The expression provided is:

[ F(x) = [[0.5]] ]

The double square brackets "[]" often represent the floor function or the greatest integer function, which rounds down to the nearest integer less than or equal to the number inside the brackets. However, this notation could also be interpreted differently depending on the context, but most commonly it is associated with the floor function.

Assuming we are using the floor function, the expression becomes:

$F(x) = \lfloor 0.5 \rfloor$

The floor function $\lfloor 0.5 \rfloor$ simply means "the greatest integer less than or equal to 0.5", which is 0.

Thus, the function simplifies to:

$F(x) = 0$

Graph:

Since the function simplifies to a constant, the graph of $F(x) = 0$ is simply a horizontal line at $y = 0$ for all $x$. It does not depend on the value of $x$, so the line stretches infinitely in both the positive and negative $x$-directions.

Domain:

The domain of this function is all real numbers since there are no restrictions on $x$. Thus, the domain is:

$\text{Domain: } (-\infty, \infty)$

Range:

The range is the set of possible $y$-values. Since the function is constant and equal to 0 for all $x$, the range is just:

$\text{Range: } \{0\}$

Summary:

• Graph: A horizontal line at $y = 0$.
• Domain: $(-\infty, \infty)$
• Range: $\{0\}$

Would you like me to generate the graph for this function?

Related Questions:

1. What would the graph look like if the function was $F(x) = \lfloor x \rfloor$?
2. How does the range of a constant function differ from that of a linear function?
3. What is the difference between the floor function and the ceiling function?
4. How would the domain change if we had a different type of function, such as a square root function?
5. Can you provide an example of a piecewise function with a restricted domain?

Tip:

The floor function always rounds down, regardless of whether the input is positive or negative. This is different from simple rounding, which rounds to the nearest integer based on the decimal.