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 $............$.

The minimum value of the function $f(x) = {x^{10}} + {x^2} + \frac{1}{{{x^{12}}}} + \frac{1}{{\left( {1\ +\ {{\sec }^{ - 1}}\ x} \right)}}$ is

Consider the function $\mathrm{f}:\left[\frac{1}{2}, 1\right] \rightarrow \mathrm{R}$ defined by $f(x)=4 \sqrt{2} x^3-3 \sqrt{2} x-1$. Consider the statements

$(I)$ The curve $y=f(x)$ intersects the $x$-axis exactly at one point

$(II)$ The curve $y=f(x)$ intersects the $x$-axis at $\mathrm{x}=\cos \frac{\pi}{12}$

Then

Let $[t]$ be the greatest integer less than or equal to $t$. Let $A$ be the set of al prime factors of $2310$ and $f: A \rightarrow \mathbb{Z}$ be the function $f(x)=\left[\log _2\left(x^2+\left[\frac{x^3}{5}\right]\right)\right]$. The number of one-to-one functions from $A$ to the range of $f$ is :

If $\mathrm{R}=\left\{(\mathrm{x}, \mathrm{y}): \mathrm{x}, \mathrm{y} \in \mathrm{Z}, \mathrm{x}^{2}+3 \mathrm{y}^{2} \leq 8\right\}$ is a relation on the set of integers $\mathrm{Z},$ then the domain of $\mathrm{R}^{-1}$ is 

Function $f(x)={\left( {1 + \frac{1}{x}} \right)^x}$ then Range of the function f (x) is