Functions

Let \(f(x)\) be a quadratic polynomial with leading coefficient 1 such that \(f(0)=p, p \neq 0\), and \(f(1)=\frac{1}{3}\). If the equations \(f(x)=0\) and \(f \circ f \circ f \circ f(x)=0\) have a common real root, then \(f(-3)\) is equal ...

The number of bijective functions \(f:\{1,3,5,7, \ldots, 99\} \rightarrow\{2,4,6,8, \ldots .100\}\), such that \(f(3) \geq f(9) \geq f(15) \geq f(21) \geq \ldots . . f(99)\), is ____________.

The number of one-one functions f : {a, b, c, d} \(\to\) {0, 1, 2, ......, 10} such
that 2f(a) \(-\) f(b) + 3f(c) + f(d) = 0 is ___________.

Let f : N \(\to\) N be a function such that f(m + n) = f(m) + f(n) for every m, n\(\in\)N. If f(6) = 18, then f(2) . f(3) is equal to :

Let S = {1, 2, 3, 4, 5, 6, 7}. Then the number of possible functions f : S \(\to\) S such that f(m . n) = f(m) . f(n) for every m, n \(\in\) S and m . n \(\in\) S is equal to _____________.

Let f : R \(\to\) R be defined as \(f(x + y) + f(x - y) = 2f(x)f(y),f\left( {{1 \over 2}} \right) = - 1\). Then, the value of \(\sum\limits_{k = 1}^{20} {{1 \over {\sin (k)\sin (k + f(k))}}}\) is equal to :

Let \(A = \{ 1,2,3,....,10\}\) and \(f:A \to A\) be defined as\(f(k) = \left\{ {\matrix{ {k + 1} & {if\,k\,is\,odd} \cr k & {if\,k\,is\,even} \cr } } \right.\)Then the number of possible functions \(g:A \to A\) such that $$gof ...

Let g : N \(\to\) N be defined asg(3n + 1) = 3n + 2,g(3n + 2) = 3n + 3,g(3n + 3) = 3n + 1, for all n \(\ge\) 0. Then which of the following statements is true?

Consider function f : A \(\to\) B and g : B \(\to\) C (A, B, C \(\subseteq\) R) such that (gof)\(-\)1 exists, then :

Let f, g : N \(\to\) N such that f(n + 1) = f(n) + f(1) \(\forall\) n\(\in\)N and g be any arbitrary function. Which of the following statements is NOT true?

A function f(x) is given by \(f(x) = {{{5^x}} \over {{5^x} + 5}}\), then the sum of the series $$f\left( {{1 \over {20}}} \right) + f\left( {{2 \over {20}}} \right) + f\left( {{3 \over {20}}} \right) + ....... + f\left( {{{39} \over {20}}} ...

Let x denote the total number of one-one functions from a set A with 3 elements to a set B with 5 elements and y denote the total number of one-one functions form the set A to the set A \(\times\) B. Then :

Let f : R → R be defined as f (x) = 2x – 1 and g : R - {1} → R be defined as g(x) =
\({{x - {1 \over 2}} \over {x - 1}}\).
Then the composition function f(g(x)) is :

If a + \(\alpha\) = 1, b + \(\beta\) = 2 and \(af(x) + \alpha f\left( {{1 \over x}} \right) = bx + {\beta \over x},x \ne 0\), then the value of the expression \({{f(x) + f\left( {{1 \over x}} \right)} \over {x + {1 \over x}}}\) is ________...

Let A = {0, 1, 2, 3, 4, 5, 6, 7}. Then the number of bijective functions f : A \(\to\) A such that f(1) + f(2) = 3 \(-\) f(3) is equal to

Let [ x ] denote the greatest integer \(\le\) x, where x \(\in\) R. If the domain of the real valued function \(f(x) = \sqrt {{{\left| {[x]} \right| - 2} \over {\left| {[x]} \right| - 3}}}\) is (\(-\) \(\infty\), a) \(]\cup\) [b, c) $$\cup...

Let \(f:R - \left\{ {{\alpha \over 6}} \right\} \to R\) be defined by \(f(x) = {{5x + 3} \over {6x - \alpha }}\). Then the value of \(\alpha\) for which (fof)(x) = x, for all \(x \in R - \left\{ {{\alpha \over 6}} \right\}\), is :

The range of the function, $$f(x) = {\log _{\sqrt 5 }}\left( {3 + \cos \left( {{{3\pi } \over 4} + x} \right) + \cos \left( {{\pi \over 4} + x} \right) + \cos \left( {{\pi \over 4} - x} \right) - \cos \left( {{{3\pi } \over 4} - x} \right...

If the functions are defined as \(f(x) = \sqrt x\) and \(g(x) = \sqrt {1 - x}\), then what is the common domain of the following functions :f + g, f \(-\) g, f/g, g/f, g \(-\) f where $$(f \pm g)(x) = f(x) \pm g(x),(f/g)x = {{f(x)} \over ...

The real valued function \(f(x) = {{\cos e{c^{ - 1}}x} \over {\sqrt {x - [x]} }}\), where [x] denotes the greatest integer less than or equal to x, is defined for all x belonging to :

Let f : R \(-\) {3} \(\to\) R \(-\) {1} be defined by f(x) = \({{x - 2} \over {x - 3}}\).Let g : R \(\to\) R be given as g(x) = 2x \(-\) 3. Then, the sum of all the values of x for which f\(-\)1(x) + g\(-\)1(x) = \({{13} \over 2}\) is e...

If f(x) and g(x) are two polynomials such that the polynomial P(x) = f(x3) + x g(x3) is divisible by x2 + x + 1, then P(1) is equal to ___________.

The inverse of \(y = {5^{\log x}}\) is :

The range of a\(\in\)R for which the function f(x) = (4a \(-\) 3)(x + loge 5) + 2(a \(-\) 7) cot\(\left( {{x \over 2}} \right)\) sin2\(\left( {{x \over 2}} \right)\), x \(\ne\) 2n\(\pi\), n\(\in\)N has critical points, is :

Let a – 2b + c = 1.
If \(f(x)=\left| {\matrix{ {x + a} & {x + 2} & {x + 1} \cr {x + b} & {x + 3} & {x + 2} \cr {x + c} & {x + 4} & {x + 3} \cr } } \right|\), then:

The inverse function of
f(x) = \({{{8^{2x}} - {8^{ - 2x}}} \over {{8^{2x}} + {8^{ - 2x}}}}\), x \(\in\) (-1, 1), is :

Let ƒ : (1, 3) \(\to\) R be a function defined by
\(f(x) = {{x\left[ x \right]} \over {1 + {x^2}}}\) , where [x] denotes the greatest
integer \(\le\) x. Then the range of ƒ is

If g(x) = x2 + x - 1 and (goƒ) (x) = 4x2 - 10x + 5, then ƒ\(\left( {{5 \over 4}} \right)\) is equal to:

If f(x + y) = f(x)f(y) and \(\sum\limits_{x = 1}^\infty {f\left( x \right)} = 2\) , x, y \(\in\) N, where N is the set of all natural number, then the
value of
\({{f\left( 4 \right)} \over {f\left( 2 \right)}}\) is :

Suppose that a function f : R \(\to\) R satisfies
f(x + y) = f(x)f(y) for all x, y \(\in\) R and f(1) = 3. If \(\sum\limits_{i = 1}^n {f(i)} = 363\) then n is equal to ________ .

For a suitably chosen real constant a, let a
function, \(f:R - \left\{ { - a} \right\} \to R\) be defined by
\(f(x) = {{a - x} \over {a + x}}\). Further suppose that for any real
number \(x \ne - a\) and \(f(x) \ne - a\),
(fof)(x) = x. ...

Let A = {a, b, c} and B = {1, 2, 3, 4}. Then the
number of elements in the set C = {f : A \(\to\) B |
2 \(\in\) f(A) and f is not one-one} is ______.

Let f : R \(\to\) R be a function which satisfies
f(x + y) = f(x) + f(y) \(\forall\) x, y \(\in\) R. If f(1) = 2 and
g(n) = \(\sum\limits_{k = 1}^{\left( {n - 1} \right)} {f\left( k \right)}\), n \(\in\) N then the value of n, for
w...

For \(x \in R - \left\{ {0,1} \right\}\), Let f1(x) = \(1\over x\), f2 (x) = 1 – x and f3 (x) = \(1 \over {1 - x}\)
be three given
functions. If a function, J(x) satisfies
(f2 o J o f1) (x) = f3 (x) then J(x) is equal to :

Let A = {x \(\in\) R : x is not a positive integer}.
Define a function \(f\) : A \(\to\)  R   as  \(f(x)\) = \({{2x} \over {x - 1}}\),
then \(f\) is :

Let \(\sum\limits_{k = 1}^{10} {f(a + k) = 16\left( {{2^{10}} - 1} \right)}\) where the function
ƒ satisfies
ƒ(x + y) = ƒ(x)ƒ(y) for all natural
numbers x, y and ƒ(1) = 2. then the natural
number 'a' is

If the function ƒ : R – {1, –1} \(\to\) A defined by
ƒ(x) = \({{{x^2}} \over {1 - {x^2}}}\) , is surjective, then A is equal to

The domain of the definition of the function
\(f(x) = {1 \over {4 - {x^2}}} + {\log _{10}}({x^3} - x)\) is

If \(f(x) = {\log _e}\left( {{{1 - x} \over {1 + x}}} \right)\), \(\left| x \right| < 1\) then \(f\left( {{{2x} \over {1 + {x^2}}}} \right)\) is equal to

Let ƒ(x) = ax
(a > 0) be written as
ƒ(x) = ƒ1
(x) + ƒ2
(x), where ƒ1
(x) is an even
function of ƒ2
(x) is an odd function. Then
ƒ1
(x + y) + ƒ1
(x – y) equals

For x \(\in\) (0, 3/2), let f(x) = \(\sqrt x\) , g(x) = tan x and h(x) = \({{1 - {x^2}} \over {1 + {x^2}}}\). If \(\phi\) (x) = ((hof)og)(x), then \(\phi \left( {{\pi \over 3}} \right)\)
is equal to :

Let fk(x) = \({1 \over k}\left( {{{\sin }^k}x + {{\cos }^k}x} \right)\) for k = 1, 2, 3, ... Then for all x \(\in\) R, the value of f4(x) \(-\) f6(x) is equal to

Let f : R \(\to\) R be defined by f(x) = \({x \over {1 + {x^2}}},x \in R\).   Then the range of f is :

Let a function f : (0, \(\infty\)) \(\to\) (0, \(\infty\)) be defined by f(x) = \(\left| {1 - {1 \over x}} \right|\). Then f is :

The number of functions f from {1, 2, 3, ...., 20} onto {1, 2, 3, ...., 20} such that f(k) is a multiple of 3,
whenever k is a multiple of 4, is :

Let N be the set of natural numbers and two functions f and g be defined as f, g : N \(\to\) N such that
f(n) = $$\left\{ {\matrix{
{{{n + 1} \over 2};} & {if\,\,n\,\,is\,\,odd} \cr
{{n \over 2};} & {if\,\,n\,\,is\,\,even} \cr

...

Let f(x) = ex – x and g(x) = x2 – x, \(\forall\) x \(\in\) R. Then the set of all x \(\in\) R, where the function h(x) = (fog) (x) is increasing, is :

Let f(x) = x2
, x \(\in\) R. For any A \(\subseteq\) R, define g (A) = { x \(\in\) R : f(x) \(\in\) A}. If S = [0,4], then which one of the
following statements is not true ?

Let f : A \(\to\) B be a function defined as f(x) = \({{x - 1} \over {x - 2}},\) Where A = R \(-\) {2} and B = R \(-\) {1}. Then   f   is :

The function f : N \(\to\) N defined by f (x) = x \(-\) 5 \(\left[ {{x \over 5}} \right],\) Where N is the set of natural numbers and [x] denotes the greatest integer less than or equal to x, is :