Let a function \( f:\mathrm{R}\to\mathrm{R} \) defined as \( f(x)=x-[x] \), (where \( \mathrm{R} \) is set of real numbers & [.] denotes greatest integer function). Then the function:
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → \( f(x) \) is neither one-one nor onto
\( f(x)=x-[x] \) is the fractional part of x, so \( 0\le f(x)\lt 1 \) for every real x.
Not one-one: different inputs give the same output, for example
\( \displaystyle f(0.5)=0.5-0=0.5 \)
\( \displaystyle f(1.5)=1.5-1=0.5 \)
Not onto: the range is [0, 1), which is not the whole co-domain R; for example, no x gives f(x) = 2.
So f is neither one-one nor onto.