Question

If $f:N\to N$ and $f\left(x\right)=x^2$ then the function is

• A. not one to one function
• B. one to one function
• C. into function
• D. none of these

Answer 1

The correct option is B one to one function

Given : $f\left(x\right)=x^2$

$\because x^2>0⟹f\left(x\right)>0$ for every $x\in N$

Let' find domain and range of $f\left(x\right)$

$x=1⟶f\left(x\right)=1$

$x=2⟶f\left(x\right)=4$

$x=3⟶f\left(x\right)=9$

$x=4⟶f\left(x\right)=16$

$x=5⟶f\left(x\right)=25$

Here we get that, no two elements of the domain has the same image and no element of co-domain is the image of more than one element in the domain.

$\therefore$ $f$ is one-one.

Function $f:X\to Y$ is onto if, for any $y\in Y$ there exist $x\in X$ such that $f\left(x\right)=y$

Let's prove that $f\left(x\right)=x^2$ is not onto.

Let's take an example $y=3\in N$

$⟹f\left(x\right)=y$ ............ by definition of onto function

$⟹x^2=3$

$⟹x=\sqrt{3}or-\sqrt{3}$ which does not belong to $N$

Hence, for $y=3\in N$ there does not exist any $x\in N$ such that $f\left(x\right)=y$.

$\therefore$ $f$ is not onto.

Hence, $f$ in only one-one.