Functions Questions (992)

The range of the function f(\theta) = \sqrt{8\sin^2 \theta + 4\cos^2 \theta - 8\sin \theta\cos \theta} is -

Domain of definition of the function \(f(x) = \dfrac{3}{4 - x^2} + \log_{10}(x^3 - x)\) is

A function \(f: R \to R\) satisfies the equation \(f(x)f(y) - f(xy) = x + y,\ \forall\ x, y \in R\) and \(f(1) > 0\), then:

Which of the following is the graph of the curve $\sqrt{|y|}=x$?

If $f(x) = \dfrac{x^3+x-2}{x^3-1}$ and $g(x) = \dfrac{x^2+x+2}{x^2+x+1}$, then ($D_f$ represents domain and $R_f$ represents range of $f(x)$)

Let \(f(x)\) and \(g(x)\) are two functions defined from \(R^+ \to R\) such that\[f(x) = \begin{cases} 1 - \sqrt{x}, & \text{if } x \text{ is rational} \\ x^2, & \text{if } x \text{ is irrational} \end{cases}\] and \[g(x) = \begin{cases} x, & \text{if } x \text{ is rational} \\ 1 - x, & \text{if } x \text{ is irrational} \end{cases}\]The composite function \(f(g(x))\) is:

Consider the graph of \(y = f(x)\) with key points \((-5,-1)\), \((-3,2)\), \((-1,1)\), \((0,3)\), \((2,3)\), \((5,-1)\) and a horizontal asymptote \(y=2\). Find the number of solution(s) of \(x\) satisfying \(f(f(x)) = 2\).

Contrapositive of the statement, "If a number is divisible by 9, then it is divisible by 3" is

If \( A \) and \( B \) are two equivalence relations defined on set \( C \), then which of the following is always true?

If $f:[-1,1]\to[0,2]$ defined by $f(x)=1+x$, then $f^{-1}$ is

Let $f$ be an even periodic function with period 2, and $f(x) = x$ for $x \in [0, 1]$. Then $f(3.14)$ equals

If $f(x)=x^2+ax+3$ and $g(x)=x+b$, where $F(x)=\lim_{n\to\infty}\dfrac{f(x)+(x^2)^n g(x)}{1+(x^2)^n}$. If $F(x)$ is continuous at $x=1$ and $x=-1$, then find the value of $(a^2+b^2)$

The negation of \(\sim s \vee (\sim r \wedge s)\) is equivalent to

If \( y = f(x) \) is onto and \( 1 , find the number of integers \([c]\) can take, i.e., find the value of \([c]_{\max}\).

Consider the statement: "P(n) : n2 − n + 41 is prime". Then which one of the following is true?

Given \( f(x) = \dfrac{2x}{x-1} \). Which of the following statements is correct?

If \(f(\ln(1+|x|)) = (1 - \ln(1+|x|))^{\frac{1}{7}}\), then \(f(f(\cos x))\) is equal to:

Which of the following is true regarding \(p \to (q \to p)\)?

Let P be the relation defined on the set of all real numbers such that \(P = \{(a, b): \sec^2 a - \tan^2 b = 1\}\). Then P is

Given \(p \rightarrow (\sim q \vee r) \equiv \sim p \wedge (\sim q \vee r)\). The truth table for \(\sim(p \wedge q) \vee r\) is given. Which option correctly represents the equivalent statement?

Consider: Statement I: \((p \wedge \sim q) \wedge (\sim p \wedge q)\) is a fallacy. Statement II: \((p \to q) \leftrightarrow (\sim q \to \sim p)\) is a tautology. (1) Statement I is true; statement II is true; statement II is a correct explanation for statement I. (2) Statement I is true; statement II is true; statement II is not a correct explanation for statement I. (3) Statement I is true; statement II is false. (4) Statement I is false; statement II is true.

Let \(S\) be a non-empty subset of \(\mathbb{R}\). Consider the following statement: \(P\): There is a rational number \(x \in S\) such that \(x > 0\). Which of the following statements is the negation of the statement \(P\)?

Consider the following two statements:Statement p: The value of sin 120° can be derived by taking \(\theta = 240°\) in the equation \(2\sin\dfrac{\theta}{2} = \sqrt{1+\sin\theta} - \sqrt{1-\sin\theta}\).Statement q: The angles A, B, C and D of any quadrilateral ABCD satisfy the equation \(\cos\left(\dfrac{1}{2}(A+C)\right) + \cos\left(\dfrac{1}{2}(B+D)\right) = 0\)Then the truth values of p and q are, respectively,

Set \(A = \{x : |x| R is

If \(A \cup B = A \cup C\) and \(A \cap B = A \cap C\), then which of the following is true?

Let \(f(x) = (x+2)^2 - 2,\ x \geq -2\). If \(g(x)\) is a function whose graph is reflection of the graph of \(y = f(x)\) in the line \(y = x\), then \(g(x)\) is equal to:

The domain of \(f(x) = \sqrt{\log\left|\frac{1}{\sin x}\right|}\) is

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

The range of the function \( f(x) = \sin^{-1}\!\left(\log_2 \dfrac{x^2}{2}\right) \) is:

If \( y = \dfrac{x^2 - x + c}{x^2 + x + c} \) and \( y \leq 1 - \dfrac{2}{1 - 2\sqrt{c}} = \dfrac{5}{3} \), find the value of \(c\).

Contrapositive of the statement"If two numbers are not equal, then their squares are not equal." is:

Let \( R \) be a relation defined on \( \mathbb{Z} \) by \( (a,b) \in R \Rightarrow a - b = nm \) where \( m \in \mathbb{Z} \) and \( m \neq 0 \). Which of the following is true?

We have \(f(x + y) = f(x) + f(y)\) for all \(x, y \in \mathbb{R}\), \(f(0) = 0\), \(f(1) = 7\). Then \(\displaystyle\sum_{r=1}^{n} f(r)\) equals:

Find all values of \(f(x)\) for which \(f(x) = x + \sqrt{x^2}\).

Given \(\displaystyle\sum_{k=1}^{10} f(a+k) = 16(2^{10}-1)\), where \(f(x+y) = f(x)f(y)\) for all \(x, y \in \mathbb{N}\) and \(f(1) = 2\). Find the value of \(a\).

A function \(f: R \to R\) satisfies the equation \(f(x)f(y) - f(xy) = x + y\), \(\forall\, x, y \in R\) and \(f(1) > 0\), then:

A survey shows that 63% of the people watch a news channel whereas, 76% watch an entertainment channel at a particular time. If \(x\%\) of the people watch both types of channels, then

Let f : R - {0} \to R be a function such that f (x) - 6f ( 1 ) = 35 - 5 . x 3x 2 If the lim x\to0 ( 1 \alphax + f (x)) = \beta; \alpha, \beta \in R , then \alpha + 2\beta is equal to

If g is the inverse of a function f and \(f'(x) = \dfrac{1}{1+x^5}\), then \(g'(x)\) is equal to

If roots of the equation \(|x-1| = 2[x] - 3\{x\}\) is \(x_i\), then find \(\left[\dfrac{\sum x_i}{5}\right]\) here (where \([x]\) & \(\{x\}\) denotes integral and fractional part of \(x\))

If $f(x) = \dfrac{2^{2x}}{2^{2x}+2}$, $x \in \mathbb{R}$, then $f\!\left(\dfrac{1}{2023}\right) + f\!\left(\dfrac{2}{2023}\right) + \cdots + f\!\left(\dfrac{2022}{2023}\right)$ is equal to

Let R1 = {(p, pn) : p prime, n ≥0, pn ≤50} and R2 = {(p, pn) : p prime, n = 0 or 1} on {1, . . . , 50}. The number of elements in R1 −R2 is:

The domain of the real-valued function \(f(x) = \frac{x-2}{(x-1)\sqrt{x^2-4}}\) is

The number of real roots of the equation 5 + |2x - 1| = 2x (2x - 2) is :

Find the inverse of \(y=5^{\log x}\).

Consider the following statements P : Suman is brilliant Q : Suman is rich R : Suman is honest The negation of the statement "Suman is brilliant and dishonest if any only if Suman is rich" can be expressed as

Let \(f\) be a function defined on the set of real numbers such that for \(x \geq 0\), \(f(x) = 3\sin x + 4\cos x\). Then \(f(x)\) at \(x = -\dfrac{11\pi}{6}\) is equal to

Let \(f : \mathbb{N} \to Y\) be a function defined as \(f(x) = 4x + 3\), where \(Y = \{y \in \mathbb{N} : y = 4x + 3 \text{ for some } x \in \mathbb{N}\}\). Show that f is invertible and its inverse is

JM Q20.

Let $S = \{1,2,3,4,5,6\}$. Then the number of one-one functions $f: S \to P(S)$, where $P(S)$ denotes the power set of $S$, such that $f(n) \subset f(m)$ where $n < m$ is ______.