Answer to Question #213802 in Algebra for Aroosha ch

Question

Find each of the function below, indicate whether the function in onto, on-to-one neither or both. If the function is not onto or nor one-to-one, give an example showing why

                               

F;R      R. f(x)=x^2

                           

Answer 1

Given Function :

$F: R\to R , F(x)=x^2$

$i)$ The function is not one-one function.

A one-one function is a function in which every element of co-domain has a distinct image in domain.

Here domain as well as co-domain is $R$

$F(x)=y=x^2$ .

Here , $y=(-x)^2=x^2$

$y=x^2\implies x=\pm \sqrt{y}$

which means an element in $y=F(x)$ has two different elements in domain , $-x$ and $x.$

Example : $F(x)=y=4$ for $x=\pm 2$ .

$ii)$

The function is not onto function.

A onto function is a function in which every element of co-domain has at least one image in domain.

Or, Range and Co-domain must be same.

Here domain as well as co-domain is $R$ .

But for any negative real number in co-domain will not have image in domain.

$y=x^2\implies x=\pm \sqrt{y}$ and $x$ is not defined if $y<0$ .

So, every element of co-domain has not an image in domain.

Example: $y=-4$ $=x^2$ has no element in domain.