Let $f & g$ be two functions both being defined from $R \to R$ as follows: $f (x) = \frac{x + 3 | x |}{2}$ and $g (x) = {\begin{matrix} x, & x < 0 \\ x^{2}, & x \geq 0 \end{matrix}$ , then which of the following is/are CORRECT?

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$f o g$ is continuous & differentiable everywhere

$f o g$ is discontinuous

$g o f$ is continuous& differentiable everywhere

$g o f$ is continuous but not differentiable

answer is D.

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Detailed Solution

$f (x) = {\begin{matrix} - x, x < 0 \\ 2 x, x \geq 0 \end{matrix}$

$N o w,, g o f (x) = {\begin{matrix} x^{2}, x < 0 \\ 4 x^{2}, x \geq 0 \end{matrix}$

Similarly, $f o g (x) = {\begin{matrix} - x, x < 0 \\ 2 x^{2}, x \geq 0 \end{matrix}$ which is continuous at $x = 0$ but not differentiable at $x = 0$

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