Let $f ^1( x )=\frac{3 x +2}{2 x +3}, x \in R -\left\{\frac{-3}{2}\right\}$ For $n \geq 2$, define $f ^{ n }( x )= f ^1 0 f ^{ n -1}( x )$. If $f ^5( x )=\frac{ ax + b }{ bx + a }, \operatorname{gcd}( a , b )=1$, then $a + b$ is equal to $............$.

If $f(x)$ be a polynomial function satisfying $f(x).f (\frac{1}{x}) = f(x) + f (\frac{1}{x})$  and $f(4) = 65$ then value of $f(6)$ is

Let $f: R \rightarrow R$ be a continuous function such that $f\left(x^2\right)=f\left(x^3\right)$ for all $x \in R$. Consider the following statements.

$I.$ $f$ is an odd function.

$II.$ $f$ is an even function.

$III$. $f$ is differentiable everywhere. Then,

If $f(x)$ and $g(x)$ are two polynomials such that the polynomial $P ( x )=f\left( x ^{3}\right)+ xg \left( x ^{3}\right)$ is divisible by $x^{2}+x+1,$ then $P(1)$ is equal to ....... .

If $f(x) = \frac{{\alpha x}}{{x + 1}},x \ne - 1$, for what value of $\alpha $ is $f(f(x)) = x$

If $f:\left\{ {1,2,3,4} \right\} \to \left\{ {1,2,3,4} \right\}$ and $y=f(x)$ be a function such that $\left| {f\left( \alpha  \right) - \alpha } \right| \leqslant 1$,for $\alpha  \in \left\{ {1,2,3,4} \right\}$ then total number of such functions are