Functions Questions (992)

Let \(f(x) = x^2\), \(x \in R\). 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?

If \(A\) is the null set, then the number of elements in the power set \(P(P(\phi))\) is

Let \(f: \mathbb{R}\setminus\{-\frac{15}{2}\} \to \mathbb{R}\setminus\{\frac{1}{2}\}\), \(f(x) = \dfrac{x+10}{2x+15}\). Determine whether \(f\) is one-one and/or onto.

Let \(f(x) = \dfrac{\sin([x]\pi)}{x^2+2x+3} + \sqrt{x(x-1)+\tfrac{1}{4}} + 2x-1\). Determine the nature of \(f:\mathbb{R}\to\mathbb{R}\).

Let R = {(a, b) : b = pq, p, q ≥3 prime} from {1, . . . , 60} to itself. The number of elements in R is:

The range of the function \(\sqrt{x-5}\) is \([z, \infty)\), where \(z\) is __________.

Let W denote the words in the English dictionary. Define the relation R by \(R = \{(x, y) \in W \times W\}\) the words x and y have at least one letter in common, then R is

Let \(P(n) = (3^{2^n} - 1)\). Then for \(n = 1\), \(P(1) = 3^2 - 1 = 9 - 1 = 8 = 2^3\), which is divisible by \(2^3\) but not by \(2^4\). Which of the following is the correct inductive conclusion?

If A = \{2, 3, 5\}, B = \{2, 5, 6\}, then (A \setminus B) \times (A \cap B) is

Let \(n\) be a positive integer and \(R = \{(a, b) \in \mathbb{Z} \times \mathbb{Z} \mid a - b = nm\text{, where } m \in \mathbb{Z} \text{ and } m \neq 0\}\). Then \(R\) is

Let \(R\) be a relation defined on the set of all natural numbers as \(R = \{(x, y) : x \in \mathbb{N}, 2x + y = 41\}\). Find the number of elements in the set range of this relation.

The statement \(p \rightarrow (q \vee r)\) is not equivalent to

Which of the following is the inverse of the proposition 'If a number is a prime then it is odd'?

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

Find the minimum number of roots of $f(x) = f\left(\frac{x+4}{x-2}\right)$.

Let \(P = \{(a, b) : \sec^2 a - \tan^2 b = 1\}\). Then \(P\) is:

Let \(f: \mathbb{R} \to \mathbb{R}\), \(f(x) = \dfrac{2x^2-5x+3}{8x^2+9x+11}\). Determine the nature of \(f\).

Piecewise functions \(f\) and \(g\) given. Which are true?(A) \((f+g)(1)=9\) (B) \((f-g)(3.5)=1\) (C) \((fg)(0)=24\) (D) \((f/g)(5)=8/3\)

Piecewise \(f\): \(x^2\) on \((0,2)\), \(2x-3\) on \([2,3)\), \(x+2\) on \([3,\infty)\). Which are correct?

●13. A ∩ B equals

Let f(x) = 8x + 3 and g(x) = x² + 4. Find 2(f∘g)(7) - (g∘f)(6).

In a class of 140 students numbered 1 to 140, all even numbered students opted Mathematics course, those whose number is divisible by 3 opted Physics course and those whose number is divisible by 5 opted Chemistry course. Then the number of students who did not opt for any of the three courses is:

The function \(f(x)\) satisfies the functional equation \(3f(x) + 2f\!\left(\dfrac{x+59}{x-1}\right) = 10x + 30\) for all real \(x \neq 1\). The value of \(f(7)\) is:

Given, \(A = \{1, \{2, 3\}\}\). Find the number of subsets of A.

The total possible subsets of sets A and B are \(2^m\) and \(2^n\) respectively. If \(2^m - 2^n = 56\) and \(2^{m-n} = \frac{3}{2}\), find \(m + n\).

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

\(\sim((\sim(\sim p)) \wedge q)\) is equal to

Let $f$ be a function such that $3f(x)+2f\!\left(\dfrac{m}{19x}\right)=5x$, $x\neq0$, where $m=\displaystyle\sum_{i=1}^9 i^2$. Then $f(5)-f(2)$ is equal to

The range of the function $f(x) = \frac{x+m}{x^2+1}$ $(m \in \mathbb{R})$ contains the interval $[0, 1]$. If $m \geq \frac{3}{k}$, then find $k$.

Given \(((p \wedge q) \vee (p \vee \sim q)) \wedge (\sim p \wedge \sim q)\) is equivalent to:

Consider two sets $A=\{x\in\mathbb{Z}:\left|(|x-3|-3)\right|\leq1\}$ and $B=\left\{x\in\mathbb{R}-\{1,2\}:\dfrac{(x-2)(x-4)}{x-1}\log_e(|x-2|)=0\right\}$. Then the number of onto functions $f:A\to B$ is equal to

Set A has m elements and Set B has n elements. If the total number of subsets of A is 112 more than the total number of subsets of B, then the value of \(m \times n\) is ___________.(JEE Main 2020)

The number of functions $f:\{1,2,3,4\} \to \{a \in \mathbb{Z}: |a| \le 8\}$ satisfying $f(n) + \dfrac{1}{n}f(n+1) = 1$, $\forall n \in \{1,2,3\}$ is

Find the values of \(x\) for which the following function is defined: \[f(x) = \sqrt{\dfrac{1}{|x-2| - (x-2)}}\]

Let \(P(x) = x^6 + ax^5 + bx^4 + cx^3 + dx^2 + ex + f\) be a polynomial such that \(P(1)=1, P(2)=2, P(3)=3, P(4)=4, P(5)=5, P(6)=6\). Find \(P(7)\).

The negation of \(q \vee \sim(p \wedge r)\) is

Compute: $\left[\frac{2103^3}{2104 \times 2105} - \frac{2104^3}{2105 \times 2106}\right]$. $[x]$ denotes greatest integer $\leq x$.

Which of the following is logically equivalent to \(\sim(\sim p \rightarrow q)\)?

Given function \(f: \mathbb{N} \to \mathbb{Z}\) such that \[f(x) = \begin{cases} \dfrac{n-1}{2}, & \text{when } n \text{ is odd} \\ -\dfrac{n}{2}, & \text{when } n \text{ is even} \end{cases}\] Which of the following is correct?

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

The logically equivalent statement of \(p \Rightarrow (q \lor r)\) is:

Let W denotes the words in the English dictionary. Define the relation R by \(R = \{(x, y) \in W \times W : \text{the words } x \text{ and } y \text{ have atleast one letter in common}\}\). Then, R is

Let the function f(x) = x² + x + sin x - cos x be defined on the interval [0, 1]. Find the odd and even extensions of f(x) in the interval [-1, 1].

Let \(P(x) = kx^3 + 2k^2x^2 + k^3\). If \((x-2)\) is a factor of \(P(x)\), find the sum of all real values of \(k\).

Let \(f(x) = \dfrac{\sin([x]\pi)}{x^2+2x+3} + \sqrt{x(x-1)+\tfrac{1}{4}} + 2x-1\). Determine the nature of \(f:\mathbb{R}\to\mathbb{R}\).

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) \(e^{-4x}(e^{2x}-1)^4\)

Find the period of \(f(x)=\min\{\sin x,|x|\}+\left\{\dfrac{x}{\pi}\right\}\)

Let \(P(x) = x^6 + ax^5 + bx^4 + cx^3 + dx^2 + ex + f\) be a polynomial such that \(P(1)=1, P(2)=2, P(3)=3, P(4)=4, P(5)=5, P(6)=6\). Find \(P(7)\).

Let \(f: \mathbb{R} \to \mathbb{R}\), \(f(x) = \dfrac{2x^2-5x+3}{8x^2+9x+11}\). Determine the nature of \(f\).

Piecewise functions \(f\) and \(g\) given. Which are true?(A) \((f+g)(1)=9\) (B) \((f-g)(3.5)=1\) (C) \((fg)(0)=24\) (D) \((f/g)(5)=8/3\)