Functions: JEE Questions & Solutions

Let the domain of the function f(x)=log₃ [log₅ (7-log₂(x²-10x+85))]+sin⁻¹ (|(3x-7)/(17-x)|) be (α,β]. Then α+β is equal to

Let [ · ] denote the greatest integer function, and let f(x) = min{√2 x, x²}. Let S = {x ∈ (-2,2): the function g(x) = |x| [x²] is…

Let f(x) = [x]² - [x+3] - 3, x ∈ ℝ, where [ · ] denotes the greatest integer function. Then

Let f and g be functions satisfying f(x+y) = f(x)f(y), f(1) = 7 g(x+y) = g(xy), g(1) = 1, for all x,y ∈ ℕ. If Σ_(x=1)ⁿ (f(x)/g(x)) = 19607,…

Consider the two sets A = {x ∈ ℤ: ||x - 3| - 3| ≤ 1} and B = {x ∈ ℝ - {1, 2}: (x-2)(x-4)/(x-1)logₑ(|x-2|) = 0} Then the number of onto…

If the domain of the function f(x)=log(10x²-17x+7)(18x²-11x+1) is (-∞,a)∪(b,c)∪(d,∞)-{e}, then 90(a+b+c+d+e) equals

Let f be a function such that 3f(x)+2f (m/19x)=5x, x≠0, where m=Σ_(i=1)⁹ i². Then f(5)-f(2) is equal to

Let f be a polynomial function such that f(x²+1)=x⁴+5x²+2, for all x∈ℝ. Then ∫₀³ f(x) dx is equal to:

Given below are two statements: Statement I: The function f:ℝ→ℝ defined by f(x)=x/(1+|x|) is one-one. Statement II: The function f:ℝ→ℝ…

Let the range of the function f(x) = 6 + 16 cos x · cos(π/3 - x) · cos(π/3 + x) · sin 3x · cos 6x, x ∈ R be [α, β]. Then the distance of…

Let f: ℝ - {0} → ℝ be a function such that f(x) - 6f(1/x) = 35/3x - 5/2. If lim_(x → 0) ( 1/(α x) + f(x)) = β, then α, β ∈ ℝ, and α + 2β is…

Let f(x) = (2^(x+2) + 16)/(2^(2x+1) + 2^(x+4) + 32). Then the value of 8 ( f( 1/15) + f( 2/15) + … + f( 59/15)) is equal to:

The function f: (-∞, ∞) → (-∞, 1), defined by f(x) = (2^x - 2^(-x))/(2^x + 2^(-x)) is:

The sum of all local minimum values of the function f(x) as defined below is: f(x)={ 1-2x if x<-1 1/3(7+2|x|) if -1 ≤ x ≤ 2 11/18(x-4)(x-5)…

If f(x) = 2^x/(2^x + √2), x ∈ ℝ, then Σ_(k=1)⁸¹ f(k/82) is equal to:

Let f: ℝ → ℝ be a function defined by f(x) = ( 2 + 3a)x² + ( (a+2)/(a-1))x + b, a ≠ 1. If f(x + y) = f(x) + f(y) + 1 - 2/7xy, then the…

Let [x] denote the greatest integer less than or equal to x. Then the domain of f(x) = sec⁻¹(2[x] + 1) is:

Let f: [0, 3] → A be defined by f(x) = 2x³ - 15x² + 36x + 7 and g: [0, ∞) → B be defined by g(x) = x/(x²⁰²⁵ + 1). If both functions are…

If the domain of the function log₅(18x - x² - 77) is (α, β) and the domain of the function log_(x-1)((2x² + 3x - 2)/(x² - 3x - 4)) is (γ,…

If the domain of the function f(x) = 1/√(10 + 3x - x²) + 1/√(x + |x|) is (a, b), then (1 + a)² + b² is equal to:

If the domain of the function f(x) = logₑ ( (2x-3)/(5+4x)) + sin⁻¹ ( (4+3x)/(2-x)) is [α, β), then α² + 4β is equal to

Let the domain of the function f(x) = log₂ log₄ log₆ (3 + 4x - x²) be (a, b). If ∫₀^(b-a) [x²] dx = p - √q - √r, p, q, r ∈ ℕ, gcd(p, q, r)…

Let f be a function such that f(x) + 3f(24/x) = 4x, x ≠ 0. Then f(3) + f(8) is equal to

If the domain of the function f(x) = log₇(1 - log₄(x² - 9x + 18)) is (α, β) ∪ (γ, δ), then α + β + γ + δ is equal to

Let f: ℝ → ℝ be a continuous function satisfying f(0) = 1 and f(2x) - f(x) = x for all x ∈ ℝ. If lim_(n → ∞) { f(x) - f( x/2ⁿ) } = G(x),…

Let the domain of the function f(x) = cos⁻¹ ( (4x + 5)/(3x - 7)) be [α, β] and the domain of g(x) = log₂ ( 2 - 6 log₂₇ ( 2x + 5)) be (γ,…

The function f: ℕ - {1} → ℕ defined by f(n) = highest prime factor of n, is:

If the domain of the function f(x) = cos⁻¹((2 - |x|)/4) + {logₑ(3 - x)}⁻¹ is [-α, β) - {γ}, then α + β + γ is equal to:

If the domain of the function f(x) = logₑ((2x + 3)/(4x² + x - 3)) + cos⁻¹((2x - 1)/(x + 2)) is (α, β], then the value of 5β - 4α is equal…

The function f(x) = (x² + 2x - 15)/(x² - 4x + 9) is:

Let the range of the function f(x) = 1/(2 + sin(3x) + cos(3x)), where x ∈ ℝ, be [a, b]. If α and β are respectively the A.M. and G.M. of a…

If f(x) = 2^2x/(2^2x + 2), x ∈ ℝ, then f(1/2023) + f(2/2023) + … + f(2022/2023) is equal to:

Let S = { θ ∈ [0, 2π): tan(π cos θ) + tan(π sin θ) = 0 }. Then Σ_(θ ∈ S) sin² (θ + π/4) is equal to:

Let f(x) = 2xⁿ + λ, where λ ∈ ℝ and n ∈ ℕ. Given that f(4) = 133 and f(5) = 255, Then the sum of all the positive integer divisors of f(3)…

The number of functions f: {1, 2, 3, 4} → {a∈ Z: |a| ≤ 8} satisfying f(n) + 1/n f(n+1) = 1, for n ∈ {1, 2, 3} is:

Let f: ℝ → ℝ be a function defined by f(x) = log_√m(√2(sin x - cos x) + m - 2), for some m, such that the range of f is [0, 2]. Then the…

The domain of f(x) = log_(x+1)(x - 2)/(e^(2logₑ x) - (2x + 3)), x ∈ ℝ, is:

Let f: ℝ → ℝ be a function such that f(x) = (x² + 2x + 1)/(x² + 1) Then

Suppose f is a function satisfying f(x + y) = f(x) + f(y) for all x, y ∈ ℕ and f(1) = 1/5. If the sum from n = 1 to m of [f(n)/n(n + 1)(n +…

Let f: ℝ → ℝ be a differentiable function that satisfies the relation f(x + y) = f(x) + f(y) - 1 for all x, y ∈ ℝ. If f'(0) = 2, then…

Suppose f: ℝ → (0, ∞) be a differentiable function such that 5f(x + y) = f(x) · f(y), ∀ x, y ∈ ℝ. If f(3) = 320, then Σ_(n=0)⁵ f(n) is…

The range of the function f(x) = √(3 - x) + √(2 + x) is:

Let A = { 1, 2, 3, 5, 8, 9 }. Then the number of possible functions f: A → A such that f(m · n) = f(m) · f(n) for every m, n ∈ A with m · n…

If the domain of the function f(x) = (⌊ x ⌋)/(1 + x²), where ⌊ x ⌋ is the greatest integer less than or equal to x, is [2, 6), then its…

Let f: ℝ \ { 2, 6 } → ℝ be the real-valued function defined as f(x) = (x² + 2x + 1)/(x² - 8x + 12). Then the range of f is:

Let f: ℝ - {0, 1} → ℝ be a function such that f(x) + f(1/(1-x)) = 1 + x. Then f(2) is equal to:

Let f(x) = x/(1 + xⁿ)^(1/n), x ∈ ℝ - {-1}, n ∈ ℕ, n > 2. If fⁿ(x) = (f ° f ° … upto n times)(x), then lim_(n → ∞) ∫₀¹ xⁿ⁻² fⁿ(x) dx is…

If the domain of the function f(x) = sec⁻¹ ( 2x/(5x + 3)) is [α, β] ∪ (γ, δ), then |3α + 10(β + γ) + 21δ| is equal to:

The number of integral solutions x of log_( x + 7/2) ( (x - 7)/(2x - 3))² ≥ 0 is:

The domain of the function f(x) = 1/√(⌊ x ⌋² - 3⌊ x ⌋ - 10), where ⌊ x ⌋ denotes the greatest integer less than or equal to x, is:

Let A = {1, 2, 3, 4, 5} and B = {1, 2, 3, 4, 5, 6}. Then the number of functions f: A → B satisfying f(1) + f(2) = f(4) - 1 is equal to.

The number of points where the curve f(x) = e^8x - e^6x - 3e^4x - e^2x + 1, x ∈ ℝ cuts the x-axis, is equal to.

Let D be the domain of the function f(x) = sin⁻¹ ( log₃ₓ ( (6 + 2 log₃ x)/(-5x))). If the range of the function g: D → ℝ defined by g(x) =…

For the differentiable function f: ℝ - {0} → ℝ, let 3f(x) + 2f(1/x) = 1/x - 10. Then | f(3) + f'(1/4) | is equal to:

If the domain of the function f(x)=logₑ(4x²+11x+6)+sin⁻¹(4x+3)+cos⁻¹ ((10x+6)/3) is (α,β), then 36|α+β| is equal to