The domain and range of greatest integer function f (x) = √ + 2 are
Answers
Answer:
Greatest integer function is defined as the largest integer less than or equal to x.
It is represented by [x] read as step
′
x
′
It is defined for all x
∴domain=(−∞,∞)
Range is all integers
Answer:
As Anshul Gupta's answer suggests, you can use graph of greatest integer function to understand what the domain and range is.
If you want a proof, go back to the basics. What is the domain? Set of points where the function is well defined. To say gif is well defined at some x, we need to say-
There are some integers smaller than x.
The set of such integers achieves it's supremum.
Based on construction/axioms of real numbers, try to see how both these hold for all real numbers.
As for the range, y is in the range of gif if there is some x in the domain of gif for which gif(x)=y. As we are concerned with finding the greatest integer, notice how the range will be a subset of integers. To say any integer y is in range, can you find a corresponding x