The function $g: \mathbb{R} \to \mathbb{R}$ is defined by $g(x) = \lfloor x \rfloor$, where $\lfloor x \rfloor$ denotes the floor function of $x$. This floor function outputs the greatest integer less than or equal to $x$.

Analysis of the Range

1. Definition: Since $\lfloor x \rfloor$ is the largest integer less than or equal to $x$, $g(x)$ will always yield integer values.
2. Possible Values of $g(x)$: For any real number $x$, $g(x)$ will output an integer, and it can produce any integer value depending on the choice of $x$.
   • For example, if $x = 2.7$, $g(x) = \lfloor 2.7 \rfloor = 2$.
   • For $x = -1.3$, $g(x) = \lfloor -1.3 \rfloor = -2$.
3. Range of $g(x)$: Since we can choose $x$ values to produce any integer as the floor function result, the range of $g(x)$ is the set of all integers.

Conclusion

The range of $g$ is: $\text{Range}(g) = \mathbb{Z}$ where $\mathbb{Z}$ denotes the set of all integers.

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Additional Questions:

1. How does the floor function differ from the ceiling function?
2. What is the range of $g(x) = \lceil x \rceil$, where $\lceil x \rceil$ is the ceiling function?
3. How does rounding affect functions like $g(x) = \lfloor x \rfloor$?
4. Can we define a similar function with only positive integer outputs?
5. How would the range change if we defined $g(x) = x^2$?

Tip:

When determining the range of a function, consider the outputs for all possible values in the domain. This helps identify the function’s limits and behavior over the real number set.