f: R R defined by f(x) = 1 + x^2 - Vidyadip

Question

$f: R \rightarrow R$ defined by $f(x) = 1 + x^2$

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Answer

$f: R \rightarrow R$ is defined as
$f(x) = 1 + x^2$
Let $\text{x}_1,\text{x}_2\in\text{R}$ such that $f(x_1) = f(x_2)$
$\Rightarrow1+\text{x}_{1}^{2}=1+\text{x}_{2}^{2}$
$\Rightarrow\text{x}_{1}^{2}=\text{x}_{2}^{2}$
$\Rightarrow\text{x}_{1}=\pm\text{x}_{2}$
$\therefore f(x_1) = f(x_2)$ does not imply that $x_1 = x_2.$
For instance,
$f(1) = f(-1) = 2$
$\therefore$ f is not one$-$one.
Consider an element $-2$ in co$-$domain $R.$
It is seen that $f(x) = 1 + x^2$ is positive for all $\text{x}\in\text{R}.$
Thus, there does not exist any $x$ in domain $R$ such that $f(x) = -2.$
$\therefore f $is not onto.
Hence$, f$ is neither one$-$one nor onto.

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