Functions

Let $\mathbb{N}$ denote the set of all positive integers. Consider the sets
\(A=\{1,2,3,4,5\} \text { and } B=\{1,2,3,4,5,6,7\} .\)
Let $S$ be the set of all functions $f: A \rightarrow B$ such that $f(2) \neq 2$ and $f(4) \neq 4$. Consid...

Let $\mathbb{R}$ denote the set of all real numbers. Let $f: \mathbb{R} \rightarrow \mathbb{R}$ and $g: \mathbb{R} \rightarrow(0,4)$ be functions defined by
\(f(x)=\log _e\left(x^2+2 x+4\right), \text { and } g(x)=\frac{4}{1+e^{-2 x}}\)
D...

Let ℝ denote the set of all real numbers. Let f: ℝ → ℝ be a function such that f(x) > 0 for all x ∈ ℝ, and f(x+y) = f(x)f(y) for all x, y ∈ ℝ.
Let the real numbers a₁, a₂, ..., a₅₀ be in an arithmetic progression. If f(a₃₁) = 64f(a₂₅), and
...

Let ℕ denote the set of all natural numbers, and ℤ denote the set of all integers. Consider the functions f: ℕ → ℤ and g: ℤ → ℕ defined by$$ f(n) = \begin{cases} \frac{(n + 1)}{2} & \text{if } n \text{ is odd,} \\ \frac{(4-n)}{2} & \text{if...

Let the function $f: \mathbb{R} \rightarrow \mathbb{R}$ be defined by

$$ f(x)=\frac{\sin x}{e^{\pi x}} \frac{\left(x^{2023}+2024 x+2025\right)}{\left(x^2-x+3\right)}+\frac{2}{e^{\pi x}} \frac{\left(x^{2023}+2024 x+2025\right)}{\left(x^2-x+...

Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a function such that $f(x+y)=f(x)+f(y)$ for all $x, y \in \mathbb{R}$, and $g: \mathbb{R} \rightarrow(0, \infty)$ be a function such that $g(x+y)=g(x) g(y)$ for all $x, y \in \mathbb{R}$. If $f\...

Let $f:[0,1] \rightarrow[0,1]$ be the function defined by $f(x)=\frac{x^3}{3}-x^2+\frac{5}{9} x+\frac{17}{36}$. Consider the square region $S=[0,1] \times[0,1]$. Let $G=\{(x, y) \in S: y>f(x)\}$ be called the green region and $R=\{(x, y) \i...

Let $S=(0,1) \cup(1,2) \cup(3,4)$ and $T=\{0,1,2,3\}$. Then which of the following statements is(are) true?

Let \(|M|\) denote the determinant of a square matrix \(M\). Let \(g:\left[0, \frac{\pi}{2}\right] \rightarrow \mathbb{R}\) be the function defined by
\(g(\theta)=\sqrt{f(\theta)-1}+\sqrt{f\left(\frac{\pi}{2}-\theta\right)-1}\)
where
$$
f...

Let the function \(f:(0,\pi ) \to R\) be defined by \(f(\theta ) = {(\sin \theta + \cos \theta )^2} + {(\sin \theta - \cos \theta )^4}\)Suppose the function f has a local minimum at \(\theta\) precisely when $$\theta \in \{ {\lambda _...

Let the function f : [0, 1] \(\to\) R be defined by\(f(x) = {{{4^x}} \over {{4^x} + 2}}\)Then the value of $$f\left( {{1 \over {40}}} \right) + f\left( {{2 \over {40}}} \right) + f\left( {{3 \over {40}}} \right) + ... + f\left( {{{39} \ov...

For a polynomial g(x) with real coefficients, let mg denote the number of distinct real roots of g(x). Suppose S is the set of polynomials with real coefficients defined by $$S = \{ {({x^2} - 1)^2}({a_0} + {a_1}x + {a_2}{x^2} + {a_3}{x^3}):...

Let f : [0, 2] \(\to\) R be the function defined by\(f(x) = (3 - \sin (2\pi x))\sin \left( {\pi x - {\pi \over 4}} \right) - \sin \left( {3\pi x + {\pi \over 4}} \right)\)If \(\alpha ,\,\beta \in [0,2]\) are such that $$\{ x \in [0,2]:...

If the function f : R \(\to\) R is defined by f(x) = |x| (x \(-\) sin x), then which of the following statements is TRUE?

Let \({E_1} = \left\{ {x \in R:x \ne 1\,and\,{x \over {x - 1}} > 0} \right\}\) and $${E_2} = \left\{ \matrix{
x \in {E_1}:{\sin ^{ - 1}}\left( {{{\log }_e}\left( {{x \over {x - 1}}} \right)} \right) \hfill \cr
is\,a\,real\,number \hfil...

Let X be a set with exactly 5 elements and Y be a set with exactly 7 elements. If \(\alpha\) is the number of one-one functions from X to Y and \(\beta\) is the number of onto functions from Y to X, then the value of $${1 \over {5!}}(\bet...

Let S = {1, 2, 3, .........., 9}. For k = 1, 2, .........., 5, let Nk be the number of subsets of S, each containing five elements out of which exactly k are odd. Then N1 + N2 + N3 + N4 + N5 =

Let \(f(x) = \sin \left( {{\pi \over 6}\sin \left( {{\pi \over 2}\sin x} \right)} \right)\) for all \(x \in R\) and g(x) = \({{\pi \over 2}\sin x}\) for all x\(\in\)R. Let \((f \circ g)(x)\) denote f(g(x)) and \((g \circ f)(x)\) denote g...

Let f1 : R \(\to\) R, f2 : [0, \(\infty\)) \(\to\) R, f3 : R \(\to\) R, and f4 : R \(\to\) [0, \(\infty\)) be defined by
$${f_1}\left( x \right) = \left\{ {\matrix{
{\left| x \right|} & {if\,x < 0,} \cr
{{e^x}} & {if\,x \g...

Let \(f:\left( { - {\pi \over 2},{\pi \over 2}} \right) \to R\) be given by \(f(x) = {[\log (\sec x + \tan x)]^3}\). Then,

For every pair of continuous function f, g : [0, 1] \(\to\) R such that max {f(x) : x \(\in\) [0, 1]} = max {g(x) : x \(\in\) [0, 1]}. The correct statement(s) is (are)

Let \(f:( - 1,1) \to R\) be such that \(f(\cos 4\theta ) = {2 \over {2 - {{\sec }^2}\theta }}\) for \(\theta \in \left( {0,{\pi \over 4}} \right) \cup \left( {{\pi \over 4},{\pi \over 2}} \right)\). Then the value(s) of $$f\left( {{1 \o...

The function \(f:[0,3] \to [1,29]\), defined by \(f(x) = 2{x^3} - 15{x^2} + 36x + 1\), is

Match the statements given in Column I with the intervals/union of intervals given in Column II :

Let \(f:(0,1) \to R\) be defined by \(f(x) = {{b - x} \over {1 - bx}}\), where b is a constant such that \(0 < b < 1\). Then

Let f(x) = x2 and g(x) = sin x for all x \(\in\) R. Then the set of all x satisfying \((f \circ g \circ g \circ f)(x) = (g \circ g \circ f)(x)\), where \((f \circ g)(x) = f(g(x))\), is

Consider the polynomial
\(f\left( x \right) = 1 + 2x + 3{x^2} + 4{x^3}.\)
Let \(s\) be the sum of all distinct real roots of \(f(x)\) and let \(t = \left| s \right|.\)
The real numbers lies in the interval

Consider the polynomial
\(f\left( x \right) = 1 + 2x + 3{x^2} + 4{x^3}.\)
Let \(s\) be the sum of all distinct real roots of \(f(x)\) and let \(t = \left| s \right|.\)
The function\(f'(x)\) is

Let $S=\{1,2,3,4\}$. The total number of unordered pairs of disjoint subsets of $S$ is equal to :

Let $f, g$ and $h$ be real valued functions defined on the interval $[0,1]$ by
$f(x)=e^{x^2}+e^{-x^2}$,
$g(x)=x e^{x^2}+e^{-x^2}$
and $h(x)=x^2 e^{x^2}+e^{-x^2}$.
If $a, b$ and $c$ denote, respectively, the absolute maximum of $f, g$ an...

If the function \(f(x) = {x^3} + {e^{x/2}}\) and \(g(x) = {f^{ - 1}}(x)\), then the value of \(g'(1)\) is _________.

If \(f''(x)=-f(x)\) and \(g(x)=f'(x)\) and \(\mathrm{F}(x)=\left(f\left(\frac{x}{2}\right)\right)^{2}+\left(g\left(\frac{x}{2}\right)\right)^{2}\) and given that \(\mathrm{F}(5)=5\), then \(\mathrm{F}(10)\) is equal to :

Find the range of value of \(t\) for which
\(2 \sin t=\frac{1-2 x+5 x^{2}}{3 x^{2}-2 x-1}, t \in\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]\)