Let $f : R \to R : f (x) = x^{2} + 3$ . Find the pre-images of each of the following under f :

(i) 19

(ii) 28

(iii) 2

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Solution

Given: $f (x) = x^{2} + 3$

(i) Let $x$ be the pre-image of 19. Then,

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

$∴$ 4 and -4 are the pre-images of 19.

(ii) Let $x$ be the pre-image of 28. Then,

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

$∴$ 5 and - 5 are the pre-images of 28.

(iii) Let $x$ be the pre-image of 2. Then,

$f (x) = 2 \Rightarrow x^{2} + 3 = 2 \Rightarrow x^{2} = - 1$

But, no real value of $x$ satisfies the equation, $x^{2} = - 1$

$∴$ 2 does not have any pre-image under f.

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