Let $f : R \to R : f (x) = x^{2} + 3$ . Find the pre-image of each of the following under f :
a) 19 b) 28 c) 2

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Solution

$f (x) = x^{2} + 3$
$a) 19$
$\Rightarrow x^{2} + 3 = 19$
$\Rightarrow x^{2} = 16$
$\Rightarrow x^{2} = \pm 4$

$b) 28$
$\Rightarrow x^{2} + 3 = 28$
$\Rightarrow x^{2} = 25$
$\Rightarrow x = \pm 5$

$c) 2$
$\Rightarrow x^{2} + 3 = 2$
$\Rightarrow x^{2} = - 1$
$\Rightarrow x = 1^{\circ} R$
$∴$ Pre image of $19$ is $\pm 4$
pre image of $28$ is $\pm 5$
and $2$ has no preimage as, $i^{\circ}$ has not belong to $R$

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