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# Give an example of a map$(iii)\quad which\; is\; neither\;one-one\;nor\;onto$

Note: This is the 3rd  part of a  3 part question, which is split as 3 separate questions here.

Toolbox:
• 1. function $:A \to B$ is one-one if $f(x)=f(y) =>x=y\qquad x,y \in A$
• 2.A function $f:A \to B$ into if for every $y \in B$ then exists $x \in$ such that $f(x)=y$
(iii) Let $f:R \to R$ defined by $f(x)=1+x^2$

Let $x_1 x_2 \in R$ such that $f(x_1)=f(x_2)$

$1+x_1^2=1+x_2^2$

$x_1^2=x_2^2$

$x_1=\pm x_2$

Since $f(x_1)=f(x_2)$ does not imply $x_1=x_2$

Hence f is not one-one

Consider an element -2 in codomain R.

We see that there does not exists any $x \in R$

f defined by $R \to R\; f(x)=1+x^2$ is not one one and not onto