Functions

156 questions · Mathematics · JEE Main

Functions

156 questions · Mathematics · JEE Main

If the domain of the function $f (x) = \frac{1}{10 + 3 x - x ^{2}} + \frac{1}{x + ∣ x ∣}$ is $(a, b)$ , then $(1 + a)^{2} + b^{2}$ is equal to :2025 · 2 Apr · Shift 2 · Q36 · MCQ
$If the domain of the function f (x) = lo g_{e} (\frac{2 x - 3}{5 + 4 x}) + sin^{- 1} (\frac{4 + 3 x}{2 - x}) is [α, β), then α^{2} + 4 β is equal to$ 2025 · 3 Apr · Shift 1 · Q26 · MCQ
Let $f$ be a function such that $f (x) + 3 f (\frac{24}{x}) = 4 x, x e q 0$ . Then $f (3) + f (8)$ is equal to2025 · 3 Apr · Shift 2 · Q34 · MCQ
If the domain of the function $f (x) = lo g_{7} (1 - lo g_{4} (x^{2} - 9 x + 18))$ is $(α, β) \cup (γ, o)$ , then $α + β + γ + \overset{o}{^}$ is equal to2025 · 3 Apr · Shift 2 · Q42 · MCQ
Let $f, g : (1, \infty) \to R$ be defined as $f (x) = \frac{2 x + 3}{5 x + 2}$ and $g (x) = \frac{2 - 3 x}{1 - x}$ . If the range of the function fog: $[2, 4] \to R$ is $[α, β]$ , then $\frac{1}{β - α}$ is…2025 · 4 Apr · Shift 1 · Q41 · MCQ
Let the domains of the functions $f (x) = lo g_{4} lo g_{3} lo g_{7} (8 - lo g_{2} (x^{2} + 4 x + 5))$ and $g (x) = sin^{- 1} (\frac{7 x + 10}{x - 2})$ be $(α, β)$ and $[γ, δ]$ , respectively. Then $α^{2} + β^{2} + γ^{2} + δ^{2}$ …2025 · 4 Apr · Shift 2 · Q27 · MCQ
If the range of the function $f (x) = \frac{5 - x}{x ^{2} - 3 x + 2}, x e q 1, 2,$ is $(- \infty, α] \cup [β, \infty)$ , then $α^{2} + β^{2}$ is equal to :2025 · 7 Apr · Shift 2 · Q31 · MCQ
Let the domain of the function $f (x) = cos^{- 1} (\frac{4 x + 5}{3 x - 7})$ be $[α, β]$ and the domain of $g (x) = lo g_{2} (2 - 6 lo g_{27} (2 x + 5))$ be $(γ, δ)$ . Then $∣7 (α + β) + 4 (γ + δ) ∣$ is…2025 · 8 Apr · Shift 2 · Q50 · Numerical
Let $A = {1, 2, 3, 4}$ and $B = {1, 4, 9, 16}$ . Then the number of many-one functions $f : A \to B$ such that $1 \in f (A)$ is equal to :2025 · 22 Jan · Shift 2 · Q41 · MCQ
Let $f (x) = lo g_{e} x$ and $g (x) = \frac{x ^{4} - 2 x ^{3} + 3 x ^{2} - 2 x + 2}{2 x ^{2} - 2 x + 1}$ . Then the domain of $f \circ g$ is2025 · 23 Jan · Shift 1 · Q29 · MCQ
Let $f (x) = \frac{2 ^{x + 2} + 16}{2 ^{2 x + 1} + 2 ^{x + 4} + 32}$ . Then the value of $8 (f (\frac{1}{15}) + f (\frac{2}{15}) + \dots + f (\frac{59}{15}))$ is equal to2025 · 24 Jan · Shift 1 · Q30 · MCQ
The function $f : (- \infty, \infty) \to (- \infty, 1)$ , defined by $f (x) = \frac{2 ^{x} - 2 ^{- x}}{2 ^{x} + 2 ^{- x}}$ is :2025 · 24 Jan · Shift 2 · Q45 · MCQ
If $f (x) = \frac{2 ^{x}}{2 ^{x} + 2}, x \in R$ , then $\sum_{k = 1}^{81} f (\frac{k}{82})$ is equal to2025 · 28 Jan · Shift 1 · Q27 · MCQ
Let $f : R \to R$ be a function defined by $f (x) = (2 + 3 a) x^{2} + (\frac{a + 2}{a - 1}) x + b, a e q 1$ . If $f (x + y) = f (x) + f (y) + 1 - \frac{2}{7} x y$ , then the value of $28 i = 1 \sum 5 ∣ f (i) ∣$ is2025 · 28 Jan · Shift 1 · Q43 · MCQ
Let $f : [0, 3] \to$ A be defined by $f (x) = 2 x^{3} - 15 x^{2} + 36 x + 7$ and $g : [0, \infty) \to B$ be defined by $g (x) = \frac{x ^{2025}}{x ^{2025} + 1}$ , If both the functions are onto and $S = {x \in Z; x \in A$ or $x \in B}$ , then $n (S)$ …2025 · 28 Jan · Shift 2 · Q44 · MCQ
If the domain of the function $lo g_{5} (18 x - x^{2} - 77)$ is $(α, β)$ and the domain of the function $lo g_{(x - 1)} (\frac{2 x ^{2} + 3 x - 2}{x ^{2} - 3 x - 4})$ is $(γ, δ)$ , then $α^{2} + β^{2} + γ^{2}$ is…2025 · 29 Jan · Shift 2 · Q29 · MCQ
Let $f : R \to R$ and $g : R \to R$ be defined as $f (x) = {lo g_{e} x, e^{- x}, x > 0 x \leq 0$ and $g (x) = {x, e^{x}, x ⩾ 0 x < 0$ …2024 · 1 Feb · Shift 1 · Q41 · MCQ
If the domain of the function $f (x) = \frac{x ^{2} - 25}{( 4 - x ^{2} )} + lo g_{10} (x^{2} + 2 x - 15)$ is $(- \infty, α) \cup [β, \infty)$ , then $α^{2} + β^{3}$ is equal to :2024 · 1 Feb · Shift 2 · Q31 · MCQ
Consider the function $f : R \to R$ defined by $f (x) = \frac{2 x}{1 + 9 x ^{2}}$ . If the composition of $f, 10 times (f \circ f \circ f \circ \dots \circ f) (x) = \frac{2 ^{10} x}{1 + 9 α x ^{2}}$ …2024 · 4 Apr · Shift 2 · Q56 · Numerical
Let $A = {1, 3, 7, 9, 11}$ and $B = {2, 4, 5, 7, 8, 10, 12}$ . Then the total number of one-one maps $f : A \to B$ , such that $f (1) + f (3) = 14$ , is :2024 · 5 Apr · Shift 1 · Q32 · MCQ
If $S = {a \in R : ∣2 a - 1∣ = 3 [a] + 2 {a}}$ , where $[t]$ denotes the greatest integer less than or equal to $t$ and ${t}$ represents the fractional part of $t$ , then $72 \sum_{a \in S} a$ is equal to .2024 · 5 Apr · Shift 1 · Q56 · Numerical
Let $f, g : R \to R$ be defined as : $f (x) = ∣ x - 1∣ and g (x) = {e^{x}, x + 1, x \geq 0 x \leq 0.$ Then the function $f (g (x))$ is2024 · 5 Apr · Shift 2 · Q32 · MCQ
The function $f (x) = \frac{x ^{2} + 2 x - 15}{x ^{2} - 4 x + 9}, x \in R$ is2024 · 6 Apr · Shift 1 · Q48 · MCQ
If the function $f (x) = (\frac{1}{x})^{2 x}; x > 0$ attains the maximum value at $x = \frac{1}{e}$ then :2024 · 6 Apr · Shift 2 · Q30 · MCQ
Let $f (x) = \frac{1}{7 - s i n 5 x}$ be a function defined on $R$ . Then the range of the function $f (x)$ is equal to :2024 · 6 Apr · Shift 2 · Q33 · MCQ
If the range of $f (θ) = \frac{s i n ^{4} θ + 3 c o s ^{2} θ}{s i n ^{4} θ + c o s ^{2} θ}, θ \in R$ is $[α, β]$ , then the sum of the infinite G.P., whose first term is 64 and the common ratio is $\frac{α}{β}$ …2024 · 8 Apr · Shift 1 · Q52 · Numerical
Let $f (x) = {- a x + a if if - a \leq x \leq 0 0 < x \leq a$ where $a > 0$ and $g (x) = (f (∣ x ∣) - ∣ f (x) ∣) /2$ …2024 · 8 Apr · Shift 2 · Q40 · MCQ
If the domain of the function $f (x) = sin^{- 1} (\frac{x - 1}{2 x + 3})$ is $R - (α, β)$ , then $12 α β$ is equal to :2024 · 9 Apr · Shift 1 · Q31 · MCQ
If a function $f$ satisfies $f (m + n) = f (m) + f (n)$ for all $m, n \in N$ and $f (1) = 1$ , then the largest natural number $λ$ such that $\sum_{k = 1}^{2022} f (λ + k) \leq (2022)^{2}$ …2024 · 9 Apr · Shift 1 · Q55 · Numerical
Let the range of the function $f (x) = \frac{1}{2 + s i n 3 x + c o s 3 x}, x \in R$ be $[a, b]$ . If $α$ and $β$ ar respectively the A.M. and the G.M. of $a$ and $b$ , then $\frac{α}{β}$ is equal to2024 · 9 Apr · Shift 2 · Q33 · MCQ
Let $A = {(x, y) : 2 x + 3 y = 23, x, y \in N}$ and $B = {x : (x, y) \in A}$ . Then the number of one-one functions from $A$ to $B$ is equal to .2024 · 9 Apr · Shift 2 · Q51 · Numerical
The function $f : N - {1} \to N$ ; defined by $f (n) =$ the highest prime factor of $n$ , is :2024 · 27 Jan · Shift 1 · Q50 · MCQ
Let $f : R - {\frac{- 1}{2}} \to R$ and $g : R - {\frac{- 5}{2}} \to R$ be defined as $f (x) = \frac{2 x + 3}{2 x + 1}$ and $g (x) = \frac{∣ x ∣ + 1}{2 x + 5}$ . Then, the domain of…2024 · 27 Jan · Shift 2 · Q40 · MCQ
If $f (x) = {2 + 2 x, 1 - \frac{x}{3}, - 1 \leq x < 0 0 \leq x \leq 3; g (x) = {- x, x, - 3 \leq x \leq 0 0 < x \leq 1$ , then range…2024 · 29 Jan · Shift 1 · Q40 · MCQ
If the domain of the function $f (x) = cos^{- 1} (\frac{2 - ∣ x ∣}{4}) + {lo g_{e} (3 - x)}^{- 1}$ is $[- α, β) - {γ}$ , then $α + β + γ$ is equal to :2024 · 30 Jan · Shift 1 · Q48 · MCQ
Let $A = {1, 2, 3, \dots, 7}$ and let $P (A)$ denote the power set of $A$ . If the number of functions $f : A \to P (A)$ such that $a \in f (a), \forall a \in A$ …2024 · 30 Jan · Shift 1 · Q59 · Numerical
If the domain of the function $f (x) = lo g_{e} (\frac{2 x + 3}{4 x ^{2} + x - 3}) + cos^{- 1} (\frac{2 x - 1}{x + 2})$ is $(α, β]$ , then the value of $5 β - 4 α$ is equal to2024 · 30 Jan · Shift 2 · Q48 · MCQ
If $f (x) = \frac{4 x + 3}{6 x - 4}, x e q \frac{2}{3}$ and $(f \circ f) (x) = g (x)$ , where $g : R - {\frac{2}{3}} \to R - {\frac{2}{3}}$ , then $(g o g o g) (4)$ is equal to2024 · 31 Jan · Shift 1 · Q36 · MCQ
Let $f (x) = 1 + sin^{2} x sin^{2} x sin^{2} x cos^{2} x 1 + cos^{2} x cos^{2} x sin 2 x sin 2 x 1 + sin 2 x, x \in [\frac{π}{6}, \frac{π}{3}]$ …2023 · 1 Feb · Shift 1 · Q31 · MCQ
Let $f : R - 0, 1 \to R$ be a function such that $f (x) + f (\frac{1}{1 - x}) = 1 + x$ . Then $f (2)$ is equal to2023 · 1 Feb · Shift 2 · Q38 · MCQ
Let the sets A and B denote the domain and range respectively of the function $f (x) = \frac{1}{⌈ x ⌉ - x}$ , where $⌈ x ⌉$ denotes the smallest integer greater than or equal to $x$ . Then among the statements (S1) : $A \cap B = (1, \infty) - N$ …2023 · 6 Apr · Shift 2 · Q36 · MCQ
Let $R = {a, b, c, d, e}$ and $S = {1, 2, 3, 4}$ . Total number of onto functions $f : R \to S$ such that $f (a) e q 1$ , is equal to …2023 · 8 Apr · Shift 2 · Q38 · Numerical
If domain of the function $lo g_{e} (\frac{6 x ^{2} + 5 x + 1}{2 x - 1}) + cos^{- 1} (\frac{2 x ^{2} - 3 x + 4}{3 x - 5})$ is $(α, β) \cup (γ, δ]$ , then…2023 · 8 Apr · Shift 2 · Q41 · Numerical
If $f (x) = \frac{( t a n 1 ^{\circ} ) x + l o g _{e} ( 123 )}{x l o g _{e} ( 1234 ) - ( t a n 1 ^{\circ} )}, x > 0$ , then the least value of $f (f (x)) + f (f (\frac{4}{x}))$ is :2023 · 10 Apr · Shift 1 · Q28 · MCQ
The domain of the function $f (x) = \frac{1}{[ x ] ^{2} - 3 [ x ] - 10}$ is : ( where $[x]$ denotes the greatest integer less than or equal to $x$ )2023 · 11 Apr · Shift 2 · Q36 · MCQ
Let $A = {1, 2, 3, 4, 5}$ and $B = {1, 2, 3, 4, 5, 6}$ . Then the number of functions $f : A \to B$ satisfying $f (1) + f (2) = f (4) - 1$ is equal to .2023 · 11 Apr · Shift 2 · Q38 · Numerical
Let $D$ be the domain of the function $f (x) = sin^{- 1} (lo g_{3 x} (\frac{6 + 2 l o g _{3} x}{- 5 x}))$ . If the range of the function $g : D \to R$ defined by $g (x) = x - [x], ([x]$ …2023 · 12 Apr · Shift 1 · Q34 · MCQ
For $x \in R$ , two real valued functions $f (x)$ and $g (x)$ are such that, $g (x) = x + 1$ and $f \circ g (x) = x + 3 - x$ . Then $f (0)$ is equal to2023 · 13 Apr · Shift 1 · Q34 · MCQ
The range of $f (x) = 4 sin^{- 1} (\frac{x ^{2}}{x ^{2} + 1})$ is2023 · 13 Apr · Shift 2 · Q33 · MCQ
Let $f (x)$ be a function such that $f (x + y) = f (x) . f (y)$ for all $x, y \in N$ . If $f (1) = 3$ and $k = 1 \sum n f (k) = 3279$ , then the value of n is2023 · 24 Jan · Shift 2 · Q23 · MCQ
If $f (x) = \frac{2 ^{2 x}}{2 ^{2 x} + 2}, x \in R$ , then $f (\frac{1}{2023}) + f (\frac{2}{2023}) + ... + f (\frac{2022}{2023})$ is equal to2023 · 24 Jan · Shift 2 · Q31 · MCQ
For some a, b, c $\in N$ , let $f (x) = a x - 3$ and $g (x) = x^{b} + c, x \in R$ . If $(f o g)^{- 1} (x) = (\frac{x - 7}{2})^{1/3}$ , then $(f o g) (a c) + (g o f) (b)$ is equal to .2023 · 25 Jan · Shift 1 · Q42 · Numerical
The number of functions $f : {1, 2, 3, 4} \to {a \in Z ∣ a ∣ \leq 8}$ satisfying $f (n) + \frac{1}{n} f (n + 1) = 1, \forall n \in {1, 2, 3}$ is2023 · 25 Jan · Shift 2 · Q24 · MCQ
Let $f : R \to R$ be a function defined by $f (x) = lo g_{m} {2 (sin x - cos x) + m - 2}$ , for some $m$ , such that the range of $f$ is [0, 2]. Then the value of $m$ is 2023 · 25 Jan · Shift 2 · Q33 · MCQ
Let $f (x) = 2 x^{n} + λ, λ \in R, n \in N$ , and $f (4) = 133, f (5) = 255$ . Then the sum of all the positive integer divisors of $(f (3) - f (2))$ is2023 · 25 Jan · Shift 2 · Q38 · MCQ
The domain of $f (x) = \frac{l o g _{(x + 1)} ( x - 2 )}{e ^{2 l o g_{e} x} - ( 2 x + 3 )}, x \in R$ is2023 · 29 Jan · Shift 1 · Q34 · MCQ
Let $f : R \to R$ be a function such that $f (x) = \frac{x ^{2} + 2 x + 1}{x ^{2} + 1}$ . Then2023 · 29 Jan · Shift 1 · Q35 · MCQ
Suppose $f$ is a function satisfying $f (x + y) = f (x) + f (y)$ for all $x, y \in N$ and $f (1) = \frac{1}{5}$ . If $n = 1 \sum m \frac{f ( n )}{n ( n + 1 ) ( n + 2 )} = \frac{1}{12}$ , then $m$ is equal to .2023 · 29 Jan · Shift 1 · Q42 · Numerical
Consider a function $f : N \to R$ , satisfying $f (1) + 2 f (2) + 3 f (3) + .... + x f (x) = x (x + 1) f (x); x \geq 2$ with $f (1) = 1$ . Then $\frac{1}{f ( 2022 )} + \frac{1}{f ( 2028 )}$ is equal to2023 · 29 Jan · Shift 2 · Q30 · MCQ
Let $S = {1, 2, 3, 4, 5, 6}$ . Then the number of one-one functions $f : S \to P (S)$ , where $P (S)$ denote the power set of $S$ , such that $f (n) \subset f (m)$ where $n \lt…2023 · 30 Jan · Shift 1 · Q37 · Numerical
The range of the function $f (x) = 3 - x + 2 + x$ is :2023 · 30 Jan · Shift 2 · Q33 · MCQ
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 .2023 · 30 Jan · Shift 2 · Q36 · Numerical
If the domain of the function $f (x) = \frac{[ x ]}{1 + x ^{2}}$ , where $[x]$ is greatest integer $\leq x$ , is $[2, 6)$ , then its range is2023 · 31 Jan · Shift 1 · Q31 · MCQ
Let $f : R - {2, 6} \to R$ be real valued function defined as $f (x) = \frac{x ^{2} + 2 x + 1}{x ^{2} - 8 x + 12}$ . Then range of $f$ is2023 · 31 Jan · Shift 2 · Q24 · MCQ
The absolute minimum value, of the function $f (x) = x^{2} - x + 1 + [x^{2} - x + 1]$ , where $[t]$ denotes the greatest integer function, in the interval $[- 1, 2]$ , is :2023 · 31 Jan · Shift 2 · Q32 · MCQ
The number of one-one functions f : {a, b, c, d} $\to$ {0, 1, 2, ......, 10} such that 2f(a) $-$ f(b) + 3f(c) + f(d) = 0 is .2022 · 24 Jun · Shift 1 · Q35 · Numerical
The total number of functions, $f : {1, 2, 3, 4} \to {1, 2, 3, 4, 5, 6}$ such that $f (1) + f (2) = f (3)$ , is equal to :2022 · 25 Jul · Shift 1 · Q21 · MCQ
The number of bijective functions $f : {1, 3, 5, 7, \dots, 99} \to {2, 4, 6, 8, \dots .100}$ , such that $f (3) \geq f (9) \geq f (15) \geq f (21) \geq \dots .. f (99)$ , is .2022 · 25 Jul · Shift 2 · Q25 · MCQ
Let $f (x)$ be a quadratic polynomial with leading coefficient 1 such that $f (0) = p, p e q 0$ , and $f (1) = \frac{1}{3}$ . If the equations $f (x) = 0$ and $f \circ f \circ f \circ f (x) = 0$ have a common real root, then $f (- 3)$ is equal to …2022 · 25 Jul · Shift 2 · Q38 · Numerical
Let f : N $\to$ R be a function such that $f (x + y) = 2 f (x) f (y)$ for natural numbers x and y. If f(1) = 2, then the value of $α$ for which $k = 1 \sum 10 f (α + k) = \frac{512}{3} (2^{20} - 1)$ holds, is :2022 · 25 Jun · Shift 1 · Q25 · MCQ
Let $f : R \to R$ and $g : R \to R$ be two functions defined by $f (x) = lo g_{e} (x^{2} + 1) - e^{- x} + 1$ and $g (x) = \frac{1 - 2 e ^{2 x}}{e ^{x}}$ . Then, for which of the following range of $α$ , the inequality $f (g (\frac{( α - 1 ) ^{2}}{3})) > f (g (α - \frac{5}{3}))$ …2022 · 25 Jun · Shift 1 · Q33 · MCQ
Let $f : R \to R$ be a function defined by $f (x) = (2 (1 - \frac{x ^{25}}{2}) (2 + x^{25}))^{\frac{1}{50}}$ . If the function $g (x) = f (f (f (x))) + f (f (x))$ , then the greatest integer less than or equal…2022 · 25 Jun · Shift 1 · Q42 · Numerical
Let $f (x) = \frac{x - 1}{x + 1}, x \in R - {0, - 1, 1}$ . If $f^{n + 1} (x) = f (f^{n} (x))$ for all n $\in$ N, then $f^{6} (6) + f^{7} (7)$ is equal to :2022 · 26 Jun · Shift 1 · Q21 · MCQ
Let f : R $\to$ R be defined as f (x) = x $-$ 1 and g : R $-$ {1, $-$ 1} $\to$ R be defined as $g (x) = \frac{x ^{2}}{x ^{2} - 1}$ . Then the function fog is :2022 · 26 Jun · Shift 2 · Q25 · MCQ
Let $f, g : N - {1} \to N$ be functions defined by $f (a) = α$ , where $α$ is the maximum of the powers of those primes $p$ such that $p^{α}$ divides $a$ , and $g (a) = a + 1$ , for all $a \in N - {1}$ …2022 · 27 Jul · Shift 1 · Q25 · MCQ
Let $f (x) = 2 x^{2} - x - 1$ and $S = {n \in Z : ∣ f (n) ∣ \leq 800}$ . Then, the value of $n \in S \sum f (n)$ is equal to .2022 · 27 Jul · Shift 1 · Q43 · Numerical
The number of functions $f$ , from the set $A = {x \in N : x^{2} - 10 x + 9 \leq 0}$ to the set $B = {n^{2} : n \in N}$ such that $f (x) \leq (x - 3)^{2} + 1$ , for every $x \in A$ …2022 · 27 Jul · Shift 2 · Q34 · Numerical
Let f : R $\to$ R be a function defined by $f (x) = \frac{2 e ^{2 x}}{e ^{2 x} + e}$ . Then $f (\frac{1}{100}) + f (\frac{2}{100}) + f (\frac{3}{100}) + ..... + f (\frac{99}{100})$ …2022 · 27 Jun · Shift 1 · Q36 · Numerical
Let S = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}. Define f : S $\to$ S as $f (n) = {2 n 2 n - 11,, i f n = 1, 2, 3, 4, 5 i f n = 6, 7, 8, 9, 10$ . Let g : S $\to$ S be a…2022 · 27 Jun · Shift 2 · Q37 · Numerical
Let $α, β$ and $γ$ be three positive real numbers. Let $f (x) = α x^{5} + β x^{3} + γ x, x \in R$ and $g : R \to R$ be such that $g (f (x)) = x$ for all $x \in R$ . If $a_{1}, a_{2}, a_{3}, \dots, a_{n}$ …2022 · 28 Jul · Shift 1 · Q36 · MCQ
For $p, q \in R$ , consider the real valued function $f (x) = (x - p)^{2} - q, x \in R$ and $q > 0$ . Let $a_{1}$ , $a_{2^{'}} a_{3}$ and $a_{4}$ …2022 · 28 Jul · Shift 1 · Q43 · Numerical
$Let f (x) = a x^{2} + b x + c be such that f (1) = 3, f (- 2) = λ and$ $f (3) = 4$ . If $f (0) + f (1) + f (- 2) + f (3) = 14$ , then $λ$ is equal to :2022 · 28 Jul · Shift 2 · Q24 · MCQ
Let a function f : N $\to$ N be defined by $f (n) = 2 n, n - 1, \frac{n + 1}{2}, n = 2, 4, 6, 8, ...... n = 3, 7, 11, 15, ...... n = 1, 5, 9, 13, ......$ then, f is2022 · 28 Jun · Shift 1 · Q24 · MCQ
Let S = {1, 2, 3, 4}. Then the number of elements in the set { f : S $\times$ S $\to$ S : f is onto and f (a, b) = f (b, a) $\geq$ a $\forall$ (a, b) $\in$ S $\times$ S } is .2022 · 28 Jun · Shift 2 · Q49 · Numerical
Let c, k $\in$ R. If $f (x) = (c + 1) x^{2} + (1 - c^{2}) x + 2 k$ and $f (x + y) = f (x) + f (y) - x y$ , for all x, y $\in$ R, then the value of $∣2 (f (1) + f (2) + f (3) + ...... + f (20)) ∣$ is equal to .2022 · 29 Jun · Shift 1 · Q39 · Numerical
Let f(x) and g(x) be two real polynomials of degree 2 and 1 respectively. If $f (g (x)) = 8 x^{2} - 2 x$ and $g (f (x)) = 4 x^{2} + 6 x + 1$ , then the value of $f (2) + g (2)$ is .2022 · 29 Jun · Shift 2 · Q43 · Numerical
The range of the function, $f (x) = lo g_{5} (3 + cos (\frac{3 π}{4} + x) + cos (\frac{π}{4} + x) + cos (\frac{π}{4} - x) - cos (\frac{3 π}{4} - x))$ …2021 · 1 Sep · Shift 2 · Q35 · MCQ
The range of a $\in$ R for which the function f(x) = (4a $-$ 3)(x + loge 5) + 2(a $-$ 7) cot $(\frac{x}{2})$ sin2 $(\frac{x}{2})$ , x $e$ 2n $π$ , n $\in$ N has critical points, is :2021 · 16 Mar · Shift 1 · Q33 · MCQ
The inverse of $y = 5^{l o g x}$ is :2021 · 17 Mar · Shift 1 · Q29 · MCQ
The real valued function $f (x) = \frac{c o s e c ^{- 1} x}{x - [ x ]}$ , where [x] denotes the greatest integer less than or equal to x, is defined for all x belonging to :2021 · 18 Mar · Shift 1 · Q34 · MCQ
If the functions are defined as $f (x) = x$ and $g (x) = 1 - x$ , then what is the common domain of the following functions : f + g, f $-$ g, f/g, g/f, g $-$ f where $(f \pm g) (x) = f (x) \pm g (x), (f / g) x = \frac{f ( x )}{g ( x )}$ 2021 · 18 Mar · Shift 1 · Q35 · MCQ
Let f : R $-$ {3} $\to$ R $-$ {1} be defined by f(x) = $\frac{x - 2}{x - 3}$ . Let g : R $\to$ R be given as g(x) = 2x $-$ 3. Then, the sum of all the values of x for which f $-$ 1(x) + g $-$ 1(x) = $\frac{13}{2}$ is equal to :2021 · 18 Mar · Shift 2 · Q35 · MCQ
If f(x) and g(x) are two polynomials such that the polynomial P(x) = f(x3) + x g(x3) is divisible by x2 + x + 1, then P(1) is equal to .2021 · 18 Mar · Shift 2 · Q38 · Numerical
Let [ x ] denote the greatest integer $\leq$ x, where x $\in$ R. If the domain of the real valued function $f (x) = \frac{∣ [ x ] ∣ - 2}{∣ [ x ] ∣ - 3}$ is ( $- \infty$ , a) $] \cup$ [b, c) $\cup$ [4, $\infty$ ),…2021 · 20 Jul · Shift 1 · Q28 · MCQ
Let $f : R - {\frac{α}{6}} \to R$ be defined by $f (x) = \frac{5 x + 3}{6 x - α}$ . Then the value of $α$ for which (fof)(x) = x, for all $x \in R - {\frac{α}{6}}$ , is :2021 · 20 Jul · Shift 2 · Q29 · MCQ
Let A = {0, 1, 2, 3, 4, 5, 6, 7}. Then the number of bijective functions f : A $\to$ A such that f(1) + f(2) = 3 $-$ f(3) is equal to2021 · 22 Jul · Shift 2 · Q38 · Numerical
Let f : R → R be defined as f (x) = 2x – 1 and g : R - {1} → R be defined as g(x) = $\frac{x - \frac{1}{2}}{x - 1}$ . Then the composition function f(g(x)) is :2021 · 24 Feb · Shift 1 · Q26 · MCQ
If a + $α$ = 1, b + $β$ = 2 and $a f (x) + α f (\frac{1}{x}) = b x + \frac{β}{x}, x e 0$ , then the value of the expression $\frac{f ( x ) + f ( \frac{1}{x} )}{x + \frac{1}{x}}$ is …2021 · 24 Feb · Shift 2 · Q37 · Numerical
Let f, g : N $\to$ N such that f(n + 1) = f(n) + f(1) $\forall$ n $\in$ N and g be any arbitrary function. Which of the following statements is NOT true?2021 · 25 Feb · Shift 1 · Q30 · MCQ
A function f(x) is given by $f (x) = \frac{5 ^{x}}{5 ^{x} + 5}$ , then the sum of the series $f (\frac{1}{20}) + f (\frac{2}{20}) + f (\frac{3}{20}) + ....... + f (\frac{39}{20})$ …2021 · 25 Feb · Shift 2 · Q25 · MCQ
Let x denote the total number of one-one functions from a set A with 3 elements to a set B with 5 elements and y denote the total number of one-one functions form the set A to the set A $\times$ B. Then :2021 · 25 Feb · Shift 2 · Q26 · MCQ
Let g : N $\to$ N be defined as g(3n + 1) = 3n + 2, g(3n + 2) = 3n + 3, g(3n + 3) = 3n + 1, for all n $\geq$ 0. Then which of the following statements is true?2021 · 25 Jul · Shift 1 · Q32 · MCQ
Consider function f : A $\to$ B and g : B $\to$ C (A, B, C $\subseteq$ R) such that (gof) $-$ 1 exists, then :2021 · 25 Jul · Shift 2 · Q36 · MCQ
Let $A = {1, 2, 3, ...., 10}$ and $f : A \to A$ be defined as $f (k) = {k + 1 k i f k i s o dd i f k i s e v e n$ Then the number of possible functions $g : A \to A$ such that…2021 · 26 Feb · Shift 2 · Q33 · MCQ
Let S = {1, 2, 3, 4, 5, 6, 7}. Then the number of possible functions f : S $\to$ S such that f(m . n) = f(m) . f(n) for every m, n $\in$ S and m . n $\in$ S is equal to .2021 · 27 Jul · Shift 1 · Q45 · Numerical
Let f : R $\to$ R be defined as $f (x + y) + f (x - y) = 2 f (x) f (y), f (\frac{1}{2}) = - 1$ . Then, the value of $k = 1 \sum 20 \frac{1}{s i n ( k ) s i n ( k + f ( k ))}$ is equal to :2021 · 27 Jul · Shift 2 · Q25 · MCQ
Let f : N $\to$ N be a function such that f(m + n) = f(m) + f(n) for every m, n $\in$ N. If f(6) = 18, then f(2) . f(3) is equal to :2021 · 31 Aug · Shift 2 · Q25 · MCQ
Let f : R $\to$ R be a function which satisfies f(x + y) = f(x) + f(y) $\forall$ x, y $\in$ R. If f(1) = 2 and g(n) = $k = 1 \sum (n - 1) f (k)$ , n $\in$ N then the value of n, for which g(n) = 20,…2020 · 2 Sep · Shift 2 · Q24 · MCQ
Let A = {a, b, c} and B = {1, 2, 3, 4}. Then the number of elements in the set C = {f : A $\to$ B | 2 $\in$ f(A) and f is not one-one} is .2020 · 5 Sep · Shift 2 · Q28 · Numerical
If f(x + y) = f(x)f(y) and $x = 1 \sum \infty f (x) = 2$ , x, y $\in$ N, where N is the set of all natural number, then the value of $\frac{f ( 4 )}{f ( 2 )}$ is :2020 · 6 Sep · Shift 1 · Q33 · MCQ
For a suitably chosen real constant a, let a function, $f : R - {- a} \to R$ be defined by $f (x) = \frac{a - x}{a + x}$ . Further suppose that for any real number $x e - a$ and $f (x) e - a$ , (fof)(x) = x. Then $f (- \frac{1}{2})$ …2020 · 6 Sep · Shift 2 · Q18 · MCQ
Suppose that a function f : R $\to$ R satisfies f(x + y) = f(x)f(y) for all x, y $\in$ R and f(1) = 3. If $i = 1 \sum n f (i) = 363$ then n is equal to .2020 · 6 Sep · Shift 2 · Q29 · Numerical
If g(x) = x2 + x - 1 and (goƒ) (x) = 4x2 - 10x + 5, then ƒ $(\frac{5}{4})$ is equal to:2020 · 7 Jan · Shift 1 · Q37 · MCQ
The inverse function of f(x) = $\frac{8 ^{2 x} - 8 ^{- 2 x}}{8 ^{2 x} + 8 ^{- 2 x}}$ , x $\in$ (-1, 1), is :2020 · 8 Jan · Shift 1 · Q26 · MCQ
Let ƒ : (1, 3) $\to$ R be a function defined by $f (x) = \frac{x [ x ]}{1 + x ^{2}}$ , where [x] denotes the greatest integer $\leq$ x. Then the range of ƒ is2020 · 8 Jan · Shift 2 · Q32 · MCQ
Let a – 2b + c = 1. If $f (x) = x + a x + b x + c x + 2 x + 3 x + 4 x + 1 x + 2 x + 3$ , then:2020 · 9 Jan · Shift 2 · Q24 · MCQ
If $f (x) = lo g_{e} (\frac{1 - x}{1 + x})$ , $∣ x ∣ < 1$ then $f (\frac{2 x}{1 + x ^{2}})$ is equal to2019 · 8 Apr · Shift 1 · Q29 · MCQ
Let ƒ(x) = ax (a > 0) be written as ƒ(x) = ƒ1 (x) + ƒ2 (x), where ƒ1 (x) is an even function of ƒ2 (x) is an odd function. Then ƒ1 (x + y) + ƒ1 (x – y) equals2019 · 8 Apr · Shift 2 · Q38 · MCQ
If the function ƒ : R – {1, –1} $\to$ A defined by ƒ(x) = $\frac{x ^{2}}{1 - x ^{2}}$ , is surjective, then A is equal to2019 · 9 Apr · Shift 1 · Q31 · MCQ
Let $k = 1 \sum 10 f (a + k) = 16 (2^{10} - 1)$ where the function ƒ satisfies ƒ(x + y) = ƒ(x)ƒ(y) for all natural numbers x, y and ƒ(1) = 2. then the natural number 'a' is2019 · 9 Apr · Shift 1 · Q35 · MCQ
The domain of the definition of the function $f (x) = \frac{1}{4 - x ^{2}} + lo g_{10} (x^{3} - x)$ is2019 · 9 Apr · Shift 2 · Q32 · MCQ
For $x \in R - {0, 1}$ , Let f1(x) = $\frac{1}{x}$ , f2 (x) = 1 – x and f3 (x) = $\frac{1}{1 - x}$ be three given functions. If a function, J(x) satisfies (f2 o J o f1) (x) = f3 (x) then J(x) is equal to :2019 · 9 Jan · Shift 1 · Q32 · MCQ
Let A = {x $\in$ R : x is not a positive integer}. Define a function $f$ : A $\to$ R as $f (x)$ = $\frac{2 x}{x - 1}$ , then $f$ is :2019 · 9 Jan · Shift 2 · Q47 · MCQ
Let f(x) = ex – x and g(x) = x2 – x, $\forall$ x $\in$ R. Then the set of all x $\in$ R, where the function h(x) = (fog) (x) is increasing, is :2019 · 10 Apr · Shift 1 · Q32 · MCQ
Let f(x) = x2 , 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 ?2019 · 10 Apr · Shift 1 · Q43 · MCQ
Let N be the set of natural numbers and two functions f and g be defined as f, g : N $\to$ N such that f(n) = ${\frac{n + 1}{2}; \frac{n}{2}; i f n i s o dd i f n i s e v e n$ …2019 · 10 Jan · Shift 2 · Q46 · MCQ
Let f : R $\to$ R be defined by f(x) = $\frac{x}{1 + x ^{2}}, x \in R$ . Then the range of f is :2019 · 11 Jan · Shift 1 · Q29 · MCQ
Let fk(x) = $\frac{1}{k} (sin^{k} x + cos^{k} x)$ for k = 1, 2, 3, ... Then for all x $\in$ R, the value of f4(x) $-$ f6(x) is equal to2019 · 11 Jan · Shift 1 · Q39 · MCQ
Let a function f : (0, $\infty$ ) $\to$ (0, $\infty$ ) be defined by f(x) = $1 - \frac{1}{x}$ . Then f is :2019 · 11 Jan · Shift 2 · Q27 · MCQ
The number of functions f from {1, 2, 3, ...., 20} onto {1, 2, 3, ...., 20} such that f(k) is a multiple of 3, whenever k is a multiple of 4, is :2019 · 11 Jan · Shift 2 · Q36 · MCQ
For x $\in$ (0, 3/2), let f(x) = $x$ , g(x) = tan x and h(x) = $\frac{1 - x ^{2}}{1 + x ^{2}}$ . If $ϕ$ (x) = ((hof)og)(x), then $ϕ (\frac{π}{3})$ is equal to :2019 · 12 Apr · Shift 1 · Q27 · MCQ
Let f : A $\to$ B be a function defined as f(x) = $\frac{x - 1}{x - 2},$ Where A = R $-$ {2} and B = R $-$ {1}. Then f is :2018 · 15 Apr · Shift 2 · Q28 · MCQ
Let f(x) = 210.x + 1 and g(x)=310.x $-$ 1. If (fog) (x) = x, then x is equal to :2017 · 8 Apr · Shift 1 · Q29 · MCQ
The function f : N $\to$ N defined by f (x) = x $-$ 5 $[\frac{x}{5}],$ Where N is the set of natural numbers and [x] denotes the greatest integer less than or equal to x, is :2017 · 9 Apr · Shift 1 · Q30 · MCQ
Let $a$ , b, c $\in R$ . If $f$ (x) = ax2 + bx + c is such that $a$ + b + c = 3 and $f$ (x + y) = $f$ (x) + $f$ (y) + xy, $\forall x, y \in R,$ then $n = 1 \sum 10 f (n)$ is equal to2017 · Shift 0 · Q25 · MCQ
The function $f : R \to [- \frac{1}{2}, \frac{1}{2}]$ defined as $f (x) = \frac{x}{1 + x ^{2}}$ , is2017 · Shift 0 · Q31 · MCQ
For x $\in$ R, x $e$ 0, Let f0(x) = $\frac{1}{1 - x}$ and fn+1 (x) = f0(fn(x)), n = 0, 1, 2, . . . . Then the value of f100(3) + f1 $(\frac{2}{3})$ + f2 $(\frac{3}{2})$ is equal to :2016 · 9 Apr · Shift 1 · Q29 · MCQ
If $f (x) + 2 f (\frac{1}{x}) = 3 x, x e q 0$ , and $S = {x \in R : f (x) = f (- x)}$ ; then $S :$ 2016 · Shift 0 · Q37 · MCQ
The domain of the function f(x) = $\frac{1}{∣ x ∣ - x}$ is2011 · Shift 0 · Q32 · MCQ
For real x, let f(x) = x3 + 5x + 1, then2009 · Shift 0 · Q28 · MCQ
Let $f (x) = (x + 1)^{2} - 1, x \geq - 1$ Statement - 1 : The set ${x : f (x) = f^{- 1} (x)} = {0, - 1}$ . Statement - 2 : $f$ is a bijection.2009 · Shift 0 · Q29 · MCQ
Let $f : N \to Y$ be a function defined as f(x) = 4x + 3 where Y = { y $\in$ N, y = 4x + 3 for some x $\in$ N }. Show that f is invertible and its inverse is2008 · Shift 0 · Q32 · MCQ
The largest interval lying in $(- \frac{π}{2}, \frac{π}{2})$ for which the function $f (x) = 4^{- x^{2}} + cos^{- 1} (\frac{x}{2} - 1) + lo g (cos x)$ , is defined, is2007 · Shift 0 · Q42 · MCQ
Let $f : (- 1, 1) \to B$ , be a function defined by $f (x) = tan^{- 1} \frac{2 x}{1 - x ^{2}}$ , then $f$ is both one-one and onto when B is the interval2005 · Shift 0 · Q60 · MCQ
A real valued function f(x) satisfies the functional equation f(x - y) = f(x)f(y) - f(a - x)f(a + y) where a is given constant and f(0) = 1, f(2a - x) is equal to2005 · Shift 0 · Q61 · MCQ
The graph of the function y = f(x) is symmetrical about the line x = 2, then2004 · Shift 0 · Q67 · MCQ
The range of the function f(x) = $^{7 - x} P_{x - 3}$ is2004 · Shift 0 · Q68 · MCQ
If $f : R \to S$ , defined by $f (x) = sin x - 3 cos x + 1$ , is onto, then the interval of $S$ is2004 · Shift 0 · Q69 · MCQ
The domain of the function $f (x) = \frac{s i n ^{- 1} ( x - 3 )}{9 - x ^{2}}$ 2004 · Shift 0 · Q70 · MCQ
The function $f (x) = lo g (x + x^{2} + 1)$ , is2003 · Shift 0 · Q66 · MCQ
A function $f$ from the set of natural numbers to integers defined by $f (n) = {\frac{n - 1}{2}, w h e n n i s o dd - \frac{n}{2}, w h e n n i s e v e n$ is2003 · Shift 0 · Q67 · MCQ
If $f : R \to R$ satisfies $f$ (x + y) = $f$ (x) + $f$ (y), for all x, y $\in$ R and $f$ (1) = 7, then $r = 1 \sum n f (r)$ is2003 · Shift 0 · Q68 · MCQ
Domain of definition of the function f(x) = $\frac{3}{4 - x ^{2}}$ + $lo g_{10} (x^{3} - x)$ , is2003 · Shift 0 · Q69 · MCQ
The domain of $sin^{- 1} [lo g_{3} (\frac{x}{3})]$ is2002 · Shift 0 · Q65 · MCQ
The period of $sin^{2} θ$ is2002 · Shift 0 · Q95 · MCQ
Which one is not periodic?2002 · Shift 0 · Q96 · MCQ