The function $y = \frac{2x - 1}{x - 2}$ $(x \neq 2)$:

Let $\alpha, \beta$ and $\gamma$ be three positive real numbers. Let $f(x) = \alpha x^5 + \beta x^3 + \gamma x, x \in \mathbb{R}$ and $g: \mathbb{R} \rightarrow \mathbb{R}$ be such that $g(f(x)) = x$ for all $x \in \mathbb{R}$. If $a_1, a_2, a_3, \dots, a_n$ are in arithmetic progression with mean zero,then the value of $f(g(\frac{1}{n} \sum_{i=1}^{n} f(a_i)))$ 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 $f(x) = \frac{x - |x|}{|x|}$,then $f(-1) = $

Let $f : R \to R$ be a function defined by $f(x) = \frac{4^x}{4^x + 2}$. What is the value of $f(\frac{1}{4}) + 2 f(\frac{1}{2}) + f(\frac{3}{4})$?

Let $f, g: R \rightarrow R$ be defined,respectively,by $f(x) = x + 1$ and $g(x) = 2x - 3$. Find $f+g$,$f-g$,and $\frac{f}{g}$.