Let $g(x) = 1 + x - [x]$ and $f(x) = \begin{cases} -1, & x < 0 \\ 0, & x = 0 \\ 1, & x > 0 \end{cases}$. Then for all $x$,$f(g(x))$ is equal to

Let $f: R \rightarrow R$ and $g: R \rightarrow R$ be the functions defined by $f(x) = \frac{x}{1+x^2}$ and $g(x) = \frac{x^2}{1+x^2}$. Then, the correct statement$(s)$ among the following is/are:

Let $f: R \rightarrow R$ be defined by $f(x)=5x^4+2$. Then

Check the injectivity and surjectivity of the following function $f: N \rightarrow N$ defined by $f(x)=x^{2}$.

The set $A$ has $4$ elements and the set $B$ has $5$ elements. Then,the number of injective mappings that can be defined from $A$ to $B$ is:

The function $f: R \rightarrow R$ is defined by $f(x)=3^{-x}$. Observe the following statements about it:
$I$. $f$ is one-one
$II$. $f$ is onto
$III$. $f$ is a decreasing function
Out of these, the true statements are: