Functions Questions - JEE

Let R = {(1, 3), (4, 2), (2, 4), (2, 3), (3, 1)} be a relation on the set A = {1, 2, 3, 4}. The relation R is

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

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

Match each function with its range: (I) $\frac{\cos^2 x+\cos x+2}{\cos^2 x+\cos x+1}$ (II) trig ratio (III) $\frac{7}{3(x^6+2x^4+3x^2+1)}$ (IV) $\log_8(x^2+2x+2)$ with (P)$(0,7/3]$ (Q)$[4/3,7/3]$ (R)$[0,1/3]$ (S)$[0,\infty)$

Let p be the statement "x is an irrational number", q be the statement "y is a transcendental number", and r be the statement "x is a rational number iff y is a transcendental number".Statement-1: r is equivalent to either q or pStatement-2: r is equivalent to $\sim (p \leftrightarrow \sim q)$.

Which of the following is correct?(i) $(A \wedge B) \wedge (\sim A \vee B) \equiv A \wedge (B \wedge (\sim A \vee B))$(ii) $(A \vee B) \wedge (\sim A \wedge B) \equiv (A \wedge \sim A) \wedge B \equiv F \wedge B \equiv F$(iii) $(A \vee B) \wedge (\sim A \vee B) \equiv B$(iv) $(A \vee B) \wedge (\sim A \vee B) \equiv B \vee (A \wedge \sim A) \equiv B \vee F \equiv F$

Let $R_1 = \{(c, a), (b, b), (a, c), (c, c), (b, c), (a, a)\}$ and $R_2 = \{(a, b), (b, a), (c, c), (c, a), (a, a), (b, b), (a, c)\}$. Which of the following is correct?

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

The domain of the real valued function $f(x)$ for which $4^{f(x)} + 4^{1-f(x)} = 4^x$ is

The logic statement $[\sim (\sim p \vee q) \wedge (p \wedge r)] \wedge (\sim p \wedge r)$ is equivalent to:

Let $W$ be the set of words in the English dictionary. Define the relation $R = \{(x, y) \in W \times W\}$ such that $x$ and $y$ have at least one letter in common. Then $R$ is:

If $p$ : 'Ram is tall' and $q$ : 'Ram is intelligent', then the statement $\sim p \vee q$ is

If $\{x\}=\frac{2}{3}$ and the nested floor/fractional-part expression equals 5, which are correct?(A) $x=\frac{14}{3}$ (B) $[x]=5$ (C) $x=\frac{17}{3}$ (D) $[x]=4$

Given $f(x) = \ln\left(\dfrac{1-x}{1+x}\right)$. Then $f\!\left(\dfrac{2x}{1+x^2}\right)$ equals:

Let S(K) = 1 + 3 + 5 + ⋯ + (2K − 1) = 3 + K2. Then which of the following is true?(1) S(1) is not true(2) Principle of Mathematical Induction cannot be used(3) S(K) ≠ S(K + 1)(4) S(K) ⟹ S(K + 1)

If $\alpha = e^{2\pi i/13}$ and $f(x) = \sum_{k=1}^{50} A_k x^k$, then find the value of $\left(\frac{1}{13}\sum_{r=0}^{12} f\left(\alpha^r\right)\right)$.

Let U be the universal set and $A \cup B \cup C = U$. Then $[(A - B) \cup (B - C) \cup (C - A)]'$ equals

Let $f_1(x) = 2^{f_2(x)}$, $f_2(x) = 2012^{f_3(x)}$, $f_3(x) = \left(\frac{1}{2013}\right)^{f_4(x)}$, where $f_4(x) = \log_{2013}(\log_x 2012)$. Find the range of $f_1(x)$.

Let $f(x)$ be such that $f(x+2) = f(x)$ and $f(-x) = f(x)$ for any real number $x$. On the interval $[2,3]$, $f(x) = x$. Then the formula of $f(x)$ given on $[-2,0]$ is:

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

(x1, y1) R (x2, y2) iff x1 ≤x2 or y1 ≤y2. Consider: (I) R is reflexive but not symmetric. (II) R is transitive. Which statement is correct?

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

It is given that the polynomial $P(x) = x^3 + ax^2 + bx + c$ has three distinct positive integer roots and $P(22) = 21$. Let $Q(x) = x^2 - 2x + 22$. It is also given that $P(Q(x))$ has no real roots then $a$ is equal to:

Least value of the expression $\frac{1}{2bx - (x^2 + b^2 + \sin^2 x)}$, $x \in [-1, 0]$, $b \in [2, 3]$ is:

Let $W$ be the set of whole numbers and $f: W \to W$ be defined by\[f(x) = \begin{cases} \left(x - 10\left[\dfrac{x}{10}\right]\right)10^{[\log_{10} x]} + f\!\left(\left[\dfrac{x}{10}\right]\right) & \text{if } x > 0 \\ 0 & \text{if } x = 0 \end{cases}\]where $[y]$ denotes the largest integer $\to y$. Then $f(7752) =$

It is given that the polynomial $P(x) = x^3 + ax^2 + bx + c$ has three distinct positive integer roots and $P(22) = 21$. Let $Q(x) = x^2 - 2x + 22$. It is also given that $P(Q(x))$ has no real roots then $a$ is equal to:

Let $f(x) = \dfrac{x}{1-x}$. If $x_0=\alpha,\ x_1=f(x_0),\ x_2=f(x_1),\ldots$ and $x_{2011} = -\dfrac{1}{2012}$, find $\alpha$.

Let $f(x, y)$ be a periodic function satisfying $f(x, y) = f(2x + 2y, 2y - 2x)$ for all $x, y$. Define $g(x) = f(2^x, 0)$. Then find the period of function $g$.

Let $f(x)=2x-\{x/\pi\}$, $g(x)=\cos x$. Find the period of $(g\circ f)(x)$.

Given $A \cap B \subseteq C$ and $A \cap B = \phi$, which of the following is NOT always true?(1) $B \cap C \neq \phi$(2) $A \subseteq C$(3) $(C \cup A) \cap (C \cup B) = C$(4) If $A = C$, then $A - C = \phi$ clearly $\phi \subseteq B$ but $A \subseteq B$ is not always true.

The only real solution to the equation $(x^2 + 100)^2 = (x^2 - 100)^2$ have how many digits in base 10 representation?

The negation of the statement "If I become a teacher, then I will open a school", is:

Let $f(x)=x+3$ for $x\in\mathbb{Q}$, $4x$ for $x\in\mathbb{R}\setminus\mathbb{Q}$; and $g(x)=\sqrt{5}+x$ for $x\in\mathbb{R}\setminus\mathbb{Q}$, $-x$ for $x\in\mathbb{Q}$. Find the nature of $(f-g)(x)$.

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

If the sets A and B are defined as$A = \{(x, y) \mid y = 1/x,\, x \neq 0,\, x \in \mathbb{R}\}$,$B = \{(x, y) \mid y = -x,\, x \in \mathbb{R}\}$. Then

Let $A = \{1, 2, 3, 5, 8, 9\}$. Then the number of possible functions $f: A \to A$ such that $f(m \cdot n) = f(m) \cdot f(n)$ for every $m, n \in A$ with $m \cdot n \in A$ is equal to ______.

If $f\left(x+\frac{1}{2}\right)+f\left(x-\frac{1}{2}\right)=f(x)$ for all $x\in\mathbb{R}$, the period is greater than:

If the sets A and B are defined as$A = \{(x, y) : y = 1/x,\; o \neq x \in \mathbb{R}\}$$B = \{(x, y) : y = -x,\; x \in \mathbb{R}\}$,Then $n(A \cap B) =$ __________.

Let f be a polynomial function such that f(3x) = f'(x) · f''(x), for all x ∈ ℝ. Then

Find the period of $f(x) = \{x\} + \{x+\frac{1}{3}\} + \{x+\frac{2}{3}\}$

Let $P = \{\theta : \sin\theta - \cos\theta = \sqrt{2}\cos\theta\}$ and $Q = \{\theta : \sin\theta + \cos\theta = \sqrt{2}\sin\theta\}$ be two sets. Then

If $f:[\frac{7}{2},\infty)\to[-\frac{9}{4},\infty)$, $f(x)=x^2-7x+10$, find $f^{-1}(x)$.

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

Which is an odd function?(A) $|x-2|+(x+2)\text{sgn}(x+2)$(B) $\frac{x}{e^x-1}+\frac{1}{2x}$(C) $\log(\sin x+\sqrt{1+\sin^2 x})$(D) $e^{-4x}(e^{2x}-1)^4$

The statement $\sim(p \leftrightarrow \sim q)$ is

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

For sets A and B, $(A \cup B)' \cup (A' \cap B)$ equals

Let $f:(0,\infty)\to(1,\infty)$, $f(x)=1+\frac{3}{2}\sqrt{x}$, and $g=f^{-1}$. Find the intersection point of $f$ and $g$.

Which of the following is inverse to itself?

Consider the statement $p$: 'New Delhi is a city'. Which of the following is not negation of $p$?

Functions Questions (992)