JAMIA 2023 — Mathematics PYQ

1. Not One-to-One (Injective)

A function is one-to-one if every $x$ has a unique $y$.

For the floor function, multiple $x$ values give the same $y$:

• $f(1.2)=\lfloor 1.2\rfloor =1$

• $f(1.8)=\lfloor 1.8\rfloor =1$

Since $1.2\ne 1.8$ but both result in $1$, it is not one-to-one.

2. Not Onto (Surjective)

A function is onto if every possible $y$ in the codomain ($\mathbb{R}$) is reached.

The output of a floor function is always an integer ($\dots ,-1,0,1,2,\dots$).

• Can you find an $x$ such that $\lfloor x\rfloor =1.5$?

• No, because $1.5$ is not an integer.

Since decimal values in the codomain are never reached, it is not onto.

Final Answer:

The floor function is neither one-to-one nor onto.