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 $…………$.

Show that the function $f: R_* \rightarrow R_*$ defined by $f(x)=\frac{1}{x}$ is one-one and onto, where $R_*$ is the set of all non-zero real numbers. Is the result true, if the domain $R_*$ is replaced by $N$ with co-domain being same as $R _*$ ?

Let $f$ be a real valued function defined by

$f(x) = sin^{-1} \left( {\frac{{\,\,1 – \,\,\left| x \right|}}{3}} \right) + cos^{-1}\left( {\frac{{\left| x \right|\,\, – \,\,3}}{5}} \right)$ .

Then domain of $f(x)$ is given by :

If ${e^x} = y + \sqrt {1 + {y^2}} $, then $y =$

Show that the function $f: N \rightarrow N$ given by $f(x)=2 x,$ is one-one but not onto.