Functions

Let f(x) = 210.x + 1 and g(x)=310.x \(-\) 1. If (fog) (x) = x, then x is equal to :

The function \(f:R \to \left[ { - {1 \over 2},{1 \over 2}} \right]\) defined as
\(f\left( x \right) = {x \over {1 + {x^2}}}\), is

Let \(a\), b, c \(\in R\). If \(f\)(x) = ax2 + bx + c is such that
\(a\) + b + c = 3 and \(f\)(x + y) = \(f\)(x) + \(f\)(y) + xy, \(\forall x,y \in R,\)
then \(\sum\limits_{n = 1}^{10} {f(n)}\) is equal to

For x \(\in\) R, x \(\ne\) 0, Let f0(x) = \({1 \over {1 - x}}\) and
fn+1 (x) = f0(fn(x)), n = 0, 1, 2, . . . .
Then the value of f100(3) + f1\(\left( {{2 \over 3}} \right)\) + f2\(\left( {{3 \over 2}} \right)\) is equal to :

If $f(x)+2 f\left(\frac{1}{x}\right)=3 x, x \neq 0$, and $\mathrm{S}=\{x \in \mathbf{R}: f(x)=f(-x)\}$; then $\mathrm{S}:$

The domain of the function f(x) = \({1 \over {\sqrt {\left| x \right| - x} }}\) is

Let \(f\left( x \right) = {\left( {x + 1} \right)^2} - 1,x \ge - 1\)
Statement - 1 : The set \(\left\{ {x:f\left( x \right) = {f^{ - 1}}\left( x \right)} \right\} = \left\{ {0, - 1} \right\}\).
Statement - 2 : \(f\) is a bijection.

For real x, let f(x) = x3 + 5x + 1, then

Let \(f:N \to Y\) be a function defined as f(x) = 4x + 3 where
Y = { y \(\in\) N, y = 4x + 3 for some x \(\in\) N }.
Show that f is invertible and its inverse is

The largest interval lying in \(\left( { - {\pi \over 2},{\pi \over 2}} \right)\) for which the function
\(f\left( x \right) = {4^{ - {x^2}}} + {\cos ^{ - 1}}\left( {{x \over 2} - 1} \right)\)\(+ \log \left( {\cos x} \right)\),
is define...

Let \(f:( - 1,1) \to B\), be a function defined by
\(f\left( x \right) = {\tan ^{ - 1}}{{2x} \over {1 - {x^2}}}\),
then \(f\) is both one-one and onto when B is the interval

A real valued function f(x) satisfies the functional equation
f(x - y) = f(x)f(y) - f(a - x)f(a + y)
where a is given constant and f(0) = 1, f(2a - x) is equal to

If \(f:R \to S\), defined by
\(f\left( x \right) = \sin x - \sqrt 3 \cos x + 1\),
is onto, then the interval of \(S\) is

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

The domain of the function
\(f\left( x \right) = {{{{\sin }^{ - 1}}\left( {x - 3} \right)} \over {\sqrt {9 - {x^2}} }}\)

The graph of the function y = f(x) is symmetrical about the line x = 2, then

Domain of definition of the function f(x) = \({3 \over {4 - {x^2}}}\) + \({\log _{10}}\left( {{x^3} - x} \right)\), is

The function \(f\left( x \right)\) \(= \log \left( {x + \sqrt {{x^2} + 1} } \right)\), is

A function \(f\) from the set of natural numbers to integers defined by
\($f\left( n \right) = \left\{ {\matrix{ {{{n - 1} \over 2},\,when\,n\,is\,odd} \cr { - {n \over 2},\,when\,n\,is\,even} \cr } } \right.\)$
is

If \(f:R \to R\) satisfies \(f\)(x + y) = \(f\)(x) + \(f\)(y), for all x, y \(\in\) R and \(f\)(1) = 7, then \(\sum\limits_{r = 1}^n {f\left( r \right)}\) is

The domain of \({\sin ^{ - 1}}\left[ {{{\log }_3}\left( {{x \over 3}} \right)} \right]\) is

Which one is not periodic?

The period of \({\sin ^2}\theta\) is