Functions Questions (992)

The equation $x^2 - 4x + [x] + 3 = x[x]$, where $[x]$ denotes the greatest integer function, has:

If \( f(x) + 2f\left(\frac{1}{x}\right) = 3x, x \neq 0, \) and S = \( \{x \in \mathbb{R} : f(x) = f(-x)\} \); then S :

Let R be a relation on \mathbb{N} \times \mathbb{N} defined by (a, b) R (c, d) if and only if ad(b - c) = bc(a - d). Then R is

Let $f(x) = a\sin x + b\sqrt[3]{x+4}$. If $f\bigl(\log_{10}(\log_3 10)\bigr) = 5$ and $f\bigl(\log_{10}(\log_3 3)\bigr) = 3$, then $f\bigl(\log_{10}(\log_3 3)\bigr)$ is

For \( x \in \mathbb{R} - \{0, 1\} \), let f_1(x) = \frac{1}{x}, f_2(x) = 1 - x and f_3(x) = \frac{1}{1 - x} \) be three given functions. If a function, J(x) satisfies \( f_2 \circ f_1 \circ f(x) = f_3(x) \) then J(x) is equal to :-

If R = {(x, y) : x, y ∈Z, x2 + 3y2 ≤8}, then the domain of R−1 is:

Let R1 = {(a, b) ∈R2 : a2 + b2 ∈Q} and R2 = {(a, b) ∈R2 : a2 + b2 /∈Q}, where Q is the set of rationals. Then:

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 R = {(a, b) : 3a −3b + √ 7 is irrational} on R. Then R is:

Which of the following functions is an odd function?

The domain of definition of the function $y = 3e^{x-1}\log(x-1)$ is

Let \( \mathbb{N} \) be the set of natural numbers and two functions f and g be defined as f,g : \( \mathbb{N} \to \mathbb{N} \) such that : \( f(n) = \frac{n + 1}{2} \) if n is odd and g(n) = n - (-1)n. The fog is :

The number of integral values of x satisfying the inequality [x-5] + [x-3] + 2[x-5] + 2[x-3] (where [.] represents greatest integer function) is

Suppose $f(x) = x^3 + \log_2\left(x + \sqrt{x^2 + 1}\right)$. For any $a, b \in \mathbb{R}$ to satisfy $f(a) + f(b) \geq 0$, the condition $a + b \geq 0$ is:

Let $f(x) = ([a]^2 - 5[a] + a)x^3 - (8[a]^2 - 5[a] + 1)x - (\tan x)\operatorname{sgn}x$, be an even function for all $x \in \{(2n+1)\frac{\pi}{2} : n \in \mathbb{Z}\}$, then sum of all possible values of $a$ is: (where $[.]$ and $\{.\}$ denotes greatest integer function and fractional part functions, respectively)

The range of $f(x) = \tan x + \frac{1}{2}\sin^{-1} x$ is:

The range of the function $f(x) = \sqrt{3-x} + \sqrt{2+x}$ is

Let f(x) = x^2, x \in \mathbb{R}. For any A \subseteq \mathbb{R}, define g(A) = \{x \in \mathbb{R} : f(x) \in A\}. If S = [0, 4], then which one of the following statements is not true ?

The domain of the function $f(x) = \frac{3}{5-x} + \log_{10}(x^2 - 3x)$ is

If the domain of the function $f(x) = \dfrac{[x]}{1+x^2}$, where $[x]$ is greatest integer $\le x$, is $(2, 6)$, then its range 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 Z be the set of integers. If A = {x ∈Z : 2(x+2)(x2−5x+6) = 1} and B = {x ∈Z : −3

Let A = {2, 3, 4, . . . , 30} and “∼” be defined on A × A by (a, b) ∼(c, d) iff ad = bc. The number of ordered pairs which are related to (4, 3) is:

Let A = {2, 3, 6, 8, 9, 11}, B = {1, 4, 5, 10, 15}. R on A × B: (a, b) R (c, d) iff 3ad −7bc is even. Then R is:

If \( f(x) + 2f\left(\frac{1}{x}\right) = 3x, x \neq 0, \) and S = \( \{x \in \mathbb{R} : f(x) = f(-x)\} \); then S :

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 :

Let $f: \mathbb{R} \to \mathbb{R}$ be a function such that $f(x) = \dfrac{x^2 + 2x + 1}{x^2 + 1}$. Then

On X = {1, . . . , 20}: R1 = {(x, y) : 2x −3y = 2} and R2 = {(x, y) : −5x + 4y = 0}. Let M, N be minimum elements to add for symmetry. M + N =?

Given below are two statements: **Statement I:** The function $f:\mathbb{R}\to\mathbb{R}$ defined by $f(x)=\dfrac{x}{1+|x|}$ is one-one. **Statement II:** The function $f:\mathbb{R}\to\mathbb{R}$ defined by $f(x)=\dfrac{x^2+4x-30}{x^2-8x+18}$ is many-one. In the light of the above statements, choose the correct answer:

If the domain of the function $f(x)=\log_{(10x^2-17x+7)}(18x^2-11x+1)$ is $(-\infty,a)\cup(b,c)\cup(d,\infty)-\{e\}$, then $90(a+b+c+d+e)$ equals:

The number of relations, defined on the set $\{a,b,c,d\}$, which are both reflexive and symmetric, is equal to:

Among the relations \( S = \left\{(a, b) : a, b \in \mathbb{R} - \{0\},\ 2 + \frac{a}{b} > 0\right\} \) and \( T = \{(a, b) : a, b \in \mathbb{R},\ a^{2} - b^{2} \in \mathbb{Z}\} \),

A function $f: \mathbb{R} \to \mathbb{R}$ has property $f(x+y) = f(x) \cdot e^{f(y)-1}$, for every $x, y \in \mathbb{R}$ then positive value of $f(4)$ is:

The function $f : [0,7] \to [0,70)$ where $f(x) = x^3 - 12x^2 + 45x$ is

Let $f: \mathbb{R} - \{2,6\} \to \mathbb{R}$ be real valued function defined as $f(x) = \dfrac{x^2+2x+1}{x^2-8x+12}$. Then range of $f$ is

Let R1 and R2 be defined on R by a R1 b ⇔ab ≥0 and a R2 b ⇔a ≥b. Then:

Let \(f\) satisfy \(f(10+x)=f(10-x)\) and \(f(20+x)=-f(20-x)\) for all \(x\in\mathbb{R}\). Which statement is correct?

Let $A=\{-2,-1,0,1,2,3,4\}$. Let $R$ be a relation on $A$ defined by $xRy$ if and only if $2x+y\leq2$. Let $l$ be the number of elements in $R$. Let $m$ and $n$ be the minimum number of elements required to be added in $R$ to make it reflexive and symmetric respectively. Then $l+m+n$ is equal to:

If $f(x)$ is symmetric about the line $x = -2$, find the values of $a$ and $b$.

Let $A=\{0,1,2,\ldots,9\}$. Let $R$ be a relation on $A$ defined by $(x,y)\in R$ if and only if $|x-y|$ is a multiple of 3. Given below are two statements: **Statement I:** $n(R)=36$. **Statement II:** $R$ is an equivalence relation. Choose the correct answer:

Let $A=\{x:|x^2-10|\leq6\}$ and $B=\{x:|x-2|>1\}$. Then

If $f(x)$ is a real valued function such that $f(x+6) - f(x+3) + f(x) = 0, \forall x \in \mathbb{R}$, then period of $f(x)$ is

Let R be a relation defined on \mathbb{N} as a R b if 2a + 3b is a multiple of 5, a, b \in \mathbb{N}. Then R is

For x \in \left(0, \frac{3}{2}\right), let f(x) = \sqrt{x}, g(x) = \tan x and h(x) = \frac{1 - x^2}{1 + x^2}. If f(x) = (hof)og(x), then \(\phi \left( \frac{\pi}{3} \right)\) is equal to :

Let $P(A)$ be defined as power set of set $A=\{1,2,3,\ldots,2025\}$. Let $R$ be a relation defined on $P(A)$ as $(B,C)\in R$ if $B$ is superset of $C$, then $R$ is

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

Let $f(x)=\frac{1}{2}\begin{vmatrix}1&\tan x&1\\-\tan x&1&\tan x\\-1&-\tan x&1\end{vmatrix}+\begin{vmatrix}\cot\frac{\pi}{2}&\sec(x+\frac{\pi}{3})&\sec(x+\frac{\pi}{12})\\\csc(x-\frac{\pi}{6})&\sin2024\pi&e^{i2024\pi}\\\csc(x-\frac{5\pi}{12})&e^{2025\pi}&\tan(2025\pi)\end{vmatrix}$ and $g(x)=\sqrt{f(x)-1}+\sqrt{f(2025\pi/2-x)-1}$ on $(0,\pi/2)$. Let $m$ be minimum of $f(x)$ and $M$ minimum of $g(x)$. Range of $h(x)=(x-m)(x-M)$ on $[0,3]$ is

Let $R=\{(a,b): a/b$ is a prime number$\}$ on first twenty natural numbers. Consider: (I) $R$ is reflexive and symmetric but not transitive; (II) Range of $R^{-1}$ has 20 elements; (III) Domain of $R^{-1}$ has 10 elements. Which statements are true?

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 :

If x^4 f\left( x \right) - \sqrt{1 - \sin 2\pi x} = |f\left( x \right)| - 2f\left( x \right), then f\left( -2 \right) equals