Question

Let N denote the set of all non-negative integers and Z denote the set of all integers. The function $f:Z\to N$ given by f(x) = |x| is :

• A. One-one but not onto
• B. Onto but not one-one
• C. Both one-one and onto
• D. Neither one-one nor onto

Answer 1

The correct option is B Onto but not one-one

$f:X\to Y$

For a function to be one one or injective, every element in the domain is the image of at most one element of it's co-domain.

In simple words, no value of $y$ must be same for $2$ or more different values of $x$.

For $f\left(x\right)=\left|x\right|$, we see that $f\left(a\right)=f(-a)$, for $a\in Z$

Hence, the function is not one one

For a function $f:X\to Y$, to be surjective,

every element $y$ in the co-domain Y must be linked with at least one element $x$ in the domain.

Every element in the co-domain of $f\left(x\right)=\left|x\right|$ is linked to at-least one element in domain.

Thus, $f\left(x\right)=\left|x\right|$ is onto but not one one.

Hence, b is correct.