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 $…………$.

Range of the function

$f(x) = \sqrt {\left| {{{\sin }^{ – 1}}\left| {\sin x} \right|} \right| – {{\cos }^{ – 1}}\left| {\cos x} \right|} $ is

If $f(x) = \frac{{{x^2} – 1}}{{{x^2} + 1}}$, for every real numbers. then the minimum value of $f$

Let $f(x ) = x^3 – 2x + 2$. If real numbers $a$, $b$ and $c$ such that $\left| {f\left( a \right)} \right| + \left| {f\left( b \right)} \right| + \left| {f\left( c \right)} \right| = 0$ then the value of ${f^2}\left( {{a^2} + \frac{2}{a}} \right) + {f^2}\left( {{b^2} + \frac{2}{b}} \right) – {f^2}\left( {{c^2} + \frac{2}{c}} \right)$ equal to

Let $S=\{1,2,3,4,5,6,7\} .$ Then the number of possible functions $f: S \rightarrow S$ such that $f(m \cdot n)=f(m) \cdot f(n)$ for every $m, n \in S$ and $m . n \in S$ is equal to $……$