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

$[t]$ denotes the greatest integer function and $[t-m] = [t] - m$ when $m \in \mathbb{Z}$. If $k = 2[2x - 1] - 1$ and $3[2x - 2] + 1 = 2[2x - 1] - 1$,then the range of $f(x) = [k + 5x]$ is

Let $f(x)$ and $g(x)$ be two continuous functions defined from $R \rightarrow R$,such that $f(x_1) > f(x_2)$ and $g(x_1) < g(x_2)$ for all $x_1 > x_2$. Then the solution set of $f(g(\alpha^2 - 2\alpha)) > f(g(3\alpha - 4))$ is

Let $f'(x) > 0$ and $g'(x) < 0$ for all $x \in R$. Then which of the following is true?

There are three kinds of liquids $X, Y, Z$. Three jars $J_1, J_2, J_3$ contain $100 \, ml$ of liquids $X, Y, Z$ respectively. An operation consists of three steps in the following order:
- Stir the liquid in $J_1$ and transfer $10 \, ml$ from $J_1$ into $J_2$.
- Stir the liquid in $J_2$ and transfer $10 \, ml$ from $J_2$ into $J_3$.
- Stir the liquid in $J_3$ and transfer $10 \, ml$ from $J_3$ into $J_1$.
After performing the operation four times,let $x, y, z$ be the amounts of $X, Y, Z$ respectively in $J_1$. Then,

Let $f : R \rightarrow R$ and $g : R \rightarrow R$ be two functions defined by $f(x)=\log _{e}(x^{2}+1)-e^{-x}+1$ and $g(x)=\frac{1-2e^{2x}}{e^{x}}$. Then,for which of the following range of $\alpha$,the inequality $f(g(\frac{(\alpha-1)^{2}}{3})) > f(g(\alpha-\frac{5}{3}))$ holds?