For x , two real valued functions f ( x ) and g ( x ) are such that, g ( x ) = x + 1 and f o g ( x ) = x + 3 - x . Then f ( 0 ) is equal to [2023]

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Solution

Q.
For $x \in ℝ$ , two real valued functions $f (x)$ and $g (x)$ are such that, $g (x) = \sqrt{x} + 1$ and $f o g (x) = x + 3 - \sqrt{x} .$ Then $f (0)$ is equal to [2023]

1 5

2 1

3 0

4 - 3

Ans.
**(*)

For $x \in R$ , $f (x)$ and $g (x)$ are real-valued functions.

$g (x) = \sqrt{x} + 1 and f o g (x) = x + 3 - \sqrt{x}$

$f o g (x) = x + 3 - \sqrt{x} \Rightarrow f (g (x)) = x + 3 - \sqrt{x}$

$= {(\sqrt{x} + 1)}^{2} - 3 (\sqrt{x} + 1) + 5$

$= {[g (x)]}^{2} - 3 [g (x)] + 5 (∵ g (x) = \sqrt{x} + 1)$

So, $f (x) = x^{2} - 3 x + 5$

$∴ f (0) = 0 - 3 \times 0 + 5 = 5$

Note:** But if we consider the domain of the composite function $f o g (x)$ , then $f (0)$ will not be defined as $g (x)$ can’t be equal to zero.

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