Let $f: R \rightarrow R, g: R \rightarrow R$ and $h: R \rightarrow R$ be differentiable functions such that $f(x)=x^3+3 x+2, g(f(x))=x$ and $h(g(x))=x$ for all $x \in R$. Then, h(0) equals ________.
Answer & explanation
We have,
$g(f(x))=x$ and $h(g(g(x)))=x$ for all $x \in R$
$\Rightarrow h(g(g(f(x))))=f(x)$ for all $x \in R$ [Replacing x by f(x)]
$\Rightarrow h(g(x))=f(x)$ for all $x \in R$ [∵ g(f(x)) = x]
$\Rightarrow h(g(f(x)))=f(f(x))$ for all $x \in R$ [Replacing x by f(x)]
$\Rightarrow h(x)=f(f(x))$ for all $x \in R$ [∵ g(f(x)) = x]
$h(x)=f(f(x))$ for all $x \in R$
∴ $h(0)=f(f(0))=f(2)=2^3+3 \times 2+2=16$ $\left[∵ f(x)=x^3+3 x+2\right]$