Functions Questions - JEE

JM Q18 — functional equation.

If a $ \in \mathbb{R} $ and the equation $ -3(x - \lfloor x \rfloor)^2 + 2(x - \lfloor x \rfloor) + a^2 = 0 $ (where $ \lfloor x \rfloor $ denotes the greatest integer $ \leq x $) has non integral real solution, then all possible values of $ a $ lie in the interval :

The complete range of values of a such that (1/2)|x| = x2 − a is satisfied for maximum number of values of x is:

The range of the function $f(x) = x^2 + \dfrac{1}{x^2 + 1}$ is:

\[f(x) = \begin{cases} x, & \text{if } x \text{ is rational} \\ 0, & \text{if } x \text{ is irrational} \end{cases}, \quad g(x) = \begin{cases} 0, & \text{if } x \text{ is rational} \\ x, & \text{if } x \text{ is irrational} \end{cases}\]\nThen, $f \circ g$ is

The following statement $(p \to q) \to [(\sim p \to q) \to q]$ is

51. If $A$ and $B$ are two equivalence relations defined on set $C$, then which of the following is always true?

Given $f(x) = \log_e(\sin x)$ and $g(x) = \sin^{-1}(e^{-x})$. If $a$ and $b$ are real numbers such that $(f \circ g)'(\alpha) = -\alpha$ gives $b = -\alpha$, and $(f \circ g)'(x) = -1 \Rightarrow (f \circ g)'(\alpha) = -1$, then find the values of $a$ and $b$.

If a + a = 1, b + b = 2 and af(x) + af$\left(\frac{1}{x}\right)$ = bx + $\frac{b}{x}$, x ≠ 0, then the value of the expression $\frac{f(x) + f\left(\frac{1}{x}\right)}{x + \frac{1}{x}}$ is

$(D): f^{-1}(x) = \frac{1}{2}\log(x + \sqrt{x^2 + 1})$

Suppose that $ f: \mathbb{R} \to \mathbb{R} $ is a continuous function and satisfies the equation $ f(x)\, f(f(x)) = 1 $ for all $ x \in \mathbb{R} $. Further, if $ f(1000) = 999 $, then which of the following options are necessarily true?$ f(500) = \dfrac{1}{500} $$ f(199) = \dfrac{1}{199} $$ f(2000) = \dfrac{1}{2000} $$ f(235) = \dfrac{1}{235} $$ f(1099) = \dfrac{1}{1099} $$ f(x) = \dfrac{1}{x} \; \forall x \in \mathbb{R} - \{0, 1000\} $No such function existsEnter the product of the number of all correct options. For example, if correct options are 2 and 3, then enter 6.

Let f : (1,3) → R be a function defined by f(x) = \frac{[x]}{1+x^2}, where [x] denotes the greatest integer \leq x. Then the range of f is

Total number of functions = $3^5$. Since each of 1, 2, 3, 4, or 5 can correspond to any of $a$, $b$, or $c$. The number of functions that corresponds to only one element of $B$ is $^3C_1 imes 1^3$ and the number of functions that correspond to almost two elements of $B$ is $^3C_2 imes 2^5$. Total number of onto functions = $3^5 - ^3C_1 imes 1^3 - (^3C_2 imes 2^5)$ (using $^3C_1 imes 1^3$ repeated twice in $^3C_2 imes 2^5$). What is the result?

If $f : \mathbb{R} \to \mathbb{R}$, $f ( x ) = ax + \cos x$ is an invertible function, then complete set of values of $a$ is:

The Boolean expression $\sim(p \wedge q) \wedge (p \vee q)$ is equivalent to

Let f: \mathbb{R} \to \mathbb{R} and f(x) = x^3 + ax^2 + bx - 8. If f(x) = 0 has three real roots & f(x) is a bijective function, then (a + b) is equal to

Let $f(x) = 4x(1-x)$, $0 \leq x \leq 1$. The number of solutions of $f(f(f(x))) = \dfrac{x}{3}$ is

Let $x$ and $y$ are real numbers satisfying $x^2 + y^2 = 4$, then find the number of integers in the range of $(x^2 - xy + y^2)$.

Let $f(x)=\dfrac{x-2}{x-3}$, $g(x)=2x-3$. Find sum of all $x$ where $f^{-1}(x)+g^{-1}(x)=\frac{13}{2}$.

141. If the range of $f(x)=\dfrac{1}{2^{\{-x\}}}-\{x\}$ is $[a,b)$ for real $x$, then the value of $'a'$ is:[Note: $\{k\}$ denotes fraction part function of $k$.]

If $f$ is a function with domain $[-3, 5]$ and $g(x) = |3x + 4|$, then the domain of $(f \circ g)(x)$ is:

Let $A = \{x \mid x^3 + x^2 - px + q = 0,\ p, q \in R\}$ and $B = \{x \mid x^2 - qx + 2 = 0,\ q \in R\}$ be the sets. If $n(A \cap B) = 2$ and $x_0 \in (A - B)$, then find the value of $|p - q + x_0|$.[Note: $n(P \cap Q)$ denotes number of common elements in set $P$ and set $Q$ and $a \in (P - Q)$ denotes elements '$a$' lies in set $P$ not in set $Q$.]

Let a function $f$ is defined as $f: \{1, 2, 3, 4, 5\} \to \{1, 2, 3, 4, 5\}$. If $f$ satisfy $f(f(x)) = f(x)$, for all $x \in \{1, 2, 3, 4\}$, then find the number of such functions.

Let $ f(x) = \dfrac{(x-1)(2x-215)}{(x-c)} $ be an onto-function, then find the greatest integral value of $ c $.

156. If $f(x) = 3x + |x|$, $g(x) = \dfrac{3x}{4} - \dfrac{|x|}{4}$, then:

141. If the range of $f(x)=\dfrac{1}{2-\{x\}}-\{x\}$ is $[a,b)$ for real $x$, then the value of $'a'$ is:[Note: $\{k\}$ denotes fraction part function of $k$.]

Let $f(x)=x^2+x$ be written as $f(x)=g(x)+h(x)$ where $g(x)$ is an odd function and $h(x)$ is an even function. Then $g(xy)+h\!\left(\dfrac{x}{y}\right)$ equals

The minimum number of elements that must be added to the relation R = \{(a, b), (b, c), (b, d)\} on the set \{a, b, c, d\} so that it is an equivalence relation, is _______.

The domain of $y(x)$ defined implicitly by $2^x + 2^y = 2$ is to be found.

Which are correct?(A) One-one self-map is onto (B) Onto self-map is one-one (C) g∘f injective ⟹ f injective (D) |A|=3,|B|=2: #functions = 8

962. If f is a quadratic polynomial, then the maximum number of roots of f(f(f(x))) = 0 is:

963. Let f : A → A where A = {1, 2, 3, 4, 5}. If f(f(x)) = f(x) for all x ∈ A, find the total number of such functions f.

Let $A=\{2,3,6,8,9,11\}$ and $B=\{1,4,5,10,15\}$. Let $R$ be a relation on $A\times B$ defined by $(a,b)\,R\,(c,d)$ if and only if $3ad-7bc$ is an even integer. Then the relation $R$ is

Find range of $f(x)=\log_{\sqrt{5}}\left(3+\cos\left(\frac{\pi}{4}+x\right)+\cos\left(\frac{\pi}{4}-x\right)+\cos\left(\frac{3\pi}{4}+x\right)-\cos\left(\frac{3\pi}{4}-x\right)\right)$.

Find the domain and range of $f(x) = \dfrac{x^2 - 3x + 2}{x^2 + x - 6}$.

Let $\sum_{k=1}^{10}f(a+k)=16(2^{10}-1)$, $f(x+y)=f(x)f(y)$, $f(1)=2$. Find $a$.

Let $A=\{x\in\mathbb{R}:[x+3]+[x+4]\leq3\}$, $B=\left\{x\in\mathbb{R}:\ 3^x\left(\displaystyle\sum_{r=1}^\infty\frac{3}{10^r}\right)^{x-3}<3^{-3x}\right\}$. Then,

Let $5f(x)+4f\!\left(\dfrac{1}{x}\right)=\dfrac{1}{x}+3$, $x>0$. Then $18\displaystyle\int_1^2 f(x)\,dx$ is equal to

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

Let $A=\{1,2,3,4,5,6,7\}$. The relation $R=\{(x,y)\in A\times A:\ x+y=7\}$ is

$|A|=5$, $|B|=2$. Number of subsets of $A\times B$ each having at least 3 and at most 6 elements is

Let $A=\{-4,-3,-2,0,1,3,4\}$ and $R=\{(a,b):\ b=|a|\ \text{or}\ b^2=a+1\}$. Minimum elements to add to make $R$ reflexive and symmetric is

On $A=\{1,2,3\}$, relations containing $(1,2)$ and $(2,3)$ which are reflexive and transitive but not symmetric number is _________.

The number of elements in $\{n\in\mathbb{N}:\ 10\leq n\leq100,\ 3^n-3\ \text{is a multiple of 7}\}$ is _______.

Let $f:\mathbb{N}\setminus\{1\}\to\mathbb{N}$, $f(n)=$highest prime factor of $n$. Determine nature of $f$.

Let f : R$\to$R be a continuous function satisfying f$(0) = 1$and f$(2x) - f$$(x) = x$for all x$\ in $R. If , then$\sum$is equal to x 10 2 limn$\to$$\infty${f$(x) - f$($)} = G(x)$G (r ) n$r=1$2

Let the domains of the functions and$g(x) = sin$be ($\alpha$,$\beta$) and [$\gamma$,$\delta$], respectively.$Then -1$$7x+10$2 f$(x) = log$log log$(8 - log$$(x + 4x + 5))$( ) 4 3 7 2$x-2$$\alpha$2 +$\beta$2 +$\gamma$2 +$\delta$2 is equal to :-

If the range of the function f$(x) = 2$$5-x$, x$\ne$1, 2, is (-$\infty$,$\alpha$]$\cup$[$\beta$,$\infty$), then$\alpha$+$\beta$is equal to : 2 2$x -3x+2$

Let f be a function such that f$(x) + 3f$( 24$) = 4x$, x$\ne$0 . Then f$(3) + f$(8) is equal to x

Let f$(x) + 2f$( 1$) = x$$2 + 5$and$2g(x) - 3g$( 1$) = x$,$x > 0$. If$\alpha$= $\int$ 2 f (x)dx , and$\beta$= $\int$ 2 g(x)dx , then the x 2 1 1 value of 9$\alpha$+$\beta$is:

Functions Questions (992)