To determine the domain of the function [tex]\( g(x) = -[x] + 3 \)[/tex], let's analyze the given function step by step.
### Function Definition
The function [tex]\( g(x) = -[x] + 3 \)[/tex] consists of two parts:
- [tex]\([x]\)[/tex] denotes the greatest integer function (also known as the floor function), which takes any real number [tex]\( x \)[/tex] and maps it to the greatest integer less than or equal to [tex]\( x \)[/tex].
- The negative sign in front of [tex]\([x]\)[/tex] indicates that we are taking the negative of the greatest integer function.
- The "+3" simply shifts the value of the output by 3 units upwards.
### Understanding the Domain
The greatest integer function [tex]\([x]\)[/tex] can accept any real number [tex]\( x \)[/tex] as its input. There is no restriction on the value of [tex]\( x \)[/tex] for this function, meaning [tex]\([x]\)[/tex] is defined for all real numbers.
Since [tex]\([x]\)[/tex] operates over all real numbers and the adjustments we make (negation and adding 3) do not impose any further restrictions, the domain of [tex]\( g(x) \)[/tex] remains the same as that of [tex]\([x]\)[/tex].
### Conclusion
The function [tex]\( g(x) = -[x] + 3 \)[/tex] is defined for all real numbers [tex]\( x \)[/tex].
Thus, the correct choice for the domain of [tex]\( g(x) \)[/tex] is:
[tex]\[
\{ x \mid x \text{ is a real number} \}
\][/tex]
