The range of the function $f(x){ = ^{7 - x}}{\kern 1pt} {P_{x - 3}}$ is

If $\,\,f(x) = \left\{ {\begin{array}{*{20}{c}}
  {3 + x;\,\,\,\,\,x \geqslant 0} \\ 
  {2 - 3x;\,\,\,\,\,x < 0} 
\end{array}} \right.$ then $\mathop {\lim }\limits_{x \to 0} f(f(x))$ is equal to -

Let $\phi (x) = (x) + {2^{\log _x^3}} - {3^{\log _x^2}}$ then

Let $f(x) = {(x + 1)^2} - 1,\;\;(x \ge - 1)$. Then the set $S = \{ x:f(x) = {f^{ - 1}}(x)\} $ is

Show that function $f : R \rightarrow\{ x \in R :-1< x <1\}$ defined by $f ( x )=\frac{x}{1+|x|^{\prime}} x \in R$ is one-one and onto function.

Let $f ( x )=2 x ^{ n }+\lambda, \lambda \in R , n \in N$, and $f (4)=133$, $f(5)=255$. Then the sum of all the positive integer divisors of $( f (3)- f (2))$ is