The total number of functions $f: \{1, 2, 3, 4\} \to \{1, 2, 3, 4, 5, 6\}$ such that $f(1) + f(2) = f(3)$ is equal to:

If $f(x) = \log_e \left( \frac{1-x}{1+x} \right)$,$|x| < 1$,then $f\left( \frac{2x}{1+x^2} \right)$ is equal to

Let $f: R \rightarrow R$ be defined by $f(x)=2x+3$. If $\alpha$ and $\beta$ are the roots of the equation $f(x^2)-2f(\frac{x}{2})-1=0$,then $\alpha^2+\beta^2=$

Let $f(x) = \frac{2^{x+2} + 16}{2^{2x+1} + 2^{x+4} + 32}$. Then the value of $8 \left( f \left( \frac{1}{15} \right) + f \left( \frac{2}{15} \right) + \dots + f \left( \frac{59}{15} \right) \right)$ is equal to

If $N$ denotes the set of all positive integers and if $f: N \rightarrow N$ is defined by $f(n) = \text{the sum of positive divisors of } n$,then $f(2^k \cdot 3)$,where $k$ is a positive integer,is

If $f: R \rightarrow R$ and $g: R \rightarrow R$ are given by $f(x)=|x|$ and $g(x)=[x]$ for each $x \in R$,then $\{x \in R: g(f(x)) \leq f(g(x))\}$ is equal to