Class 12 Relations And Functions · Very Long Answer5 Markstypes_of_functions

Class 12 Maths · Relations And Functions · Very Long Answer #190

Is the greatest-integer function f:R→R, f(x)=⌊x⌋, one-one or onto? Give reasons.

✅ Step-by-Step Solution

Final Answer: Neither one-one nor onto; range Z.

The greatest integer ⌊x⌋ is the unique integer n satisfying n≤x<n+1.
The distinct inputs 2 and 5/2 both have image 2, so injectivity fails.
Every output is an integer; in particular the codomain element 1/2 cannot be an output, so surjectivity onto R fails.
Conversely each integer n is attained at x=n, establishing that the range is exactly Z.
The function is neither one-one nor onto R, with range Z.

[1 mark] The greatest integer ⌊x⌋ is the unique integer n satisfying n≤x<n+1.
[1 mark] The distinct inputs 2 and 5/2 both have image 2, so injectivity fails.
[1 mark] Every output is an integer; in particular the codomain element 1/2 cannot be an output, so surjectivity onto R fails.
[1 mark] Conversely each integer n is attained at x=n, establishing that the range is exactly Z.
[1 mark] The function is neither one-one nor onto R, with range Z.

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