List of top Mathematics Questions on Functions

Statement 1 : The function \(f:\mathbb{R}\to\mathbb{R}\) defined by \[ f(x)=\frac{x}{1+|x|} \] is one–one.
Statement 2 : The function \(f:\mathbb{R}\to\mathbb{R}\) defined by \[ f(x)=\frac{x^2+4x-30}{x^2-8x+18} \] is many–one.
Which of the following is correct?
If \( y = \operatorname{sgn}(\sin x) + \operatorname{sgn}(\cos x) + \operatorname{sgn}(\tan x) + \operatorname{sgn}(\cot x) \), where \(\operatorname{sgn}(p)\) denotes the signum function of \(p\), then the sum of elements in the range of \(y\) is:
If domain of \(f(x) = \sin^{-1}\left(\frac{5-x}{2x+3}\right) + \frac{1}{\log_{e}(10-x)}\) is \((-\infty, \alpha] \cup (\beta, \gamma) - \{\delta\}\) then value of \(6(\alpha + \beta + \gamma + \delta)\) is equal to :
If $ f(x) = 2x^2 - 3x + 5 $, find $ f(3) $.
Let \( f: \mathbb{R} \to \mathbb{R} \) be a function defined by \( f(x) = \left( 2 + 3a \right)x^2 + \left( \frac{a+2}{a-1} \right)x + b, a \neq 1 \). If \[ f(x + y) = f(x) + f(y) + 1 - \frac{2}{7}xy, \] then the value of \( 28 \sum_{i=1}^5 f(i) \) is:
Let the function $ f(x) = \sqrt{\log_e(1 - x^2)} $. Then the domain of $ f(x) $ is:
Given the function \( F(x) = |\sin(3x)| - \cos(3x) \), for \( \frac{\pi}{6} \leq x \leq \frac{\pi}{3} \), find \( f' \left( \frac{\pi}{4} \right) \).
Given the function \( h(x) = f(g(x)) \), where \( f(x) = f'(x) = 3 \), and \( g(x) = 9 \), find \( g'(3) \), \( f'(3) \), and \( h'(3) \).
Find the domain of the composite function \( f \circ g(x) \) where \( f(x) = \log(5x) \) and \( g(x) = \cos(x) \).
If \( f(x) = \sqrt{x - 3} + 4 \sqrt{5 - x} \), find the domain of \( f(x) \).
Find the domain of the function: $ f(x) = \sqrt{7 - 11x} $
If $ f(x) = \log 3 - \sin x $, $ y = f(f(x)) $, find $ y(0) $.
Find the range of $ f(x) = \sqrt{x^2 + 4x + 4} $.
If $ f(x) = 3x^2 + 5x - 7 $, find $ f(2) $.
In an arithmetic progression, if \( S_{40} = 1030 \) and \( S_{12} = 57 \), then \( S_{30} - S_{10} \) is equal to:
Let \( f: \mathbb{R} \to \mathbb{R} \) be a function defined by \( f(x) = \left( 2 + 3a \right)x^2 + \left( \frac{a+2}{a-1} \right)x + b, a \neq 1 \). If \[ f(x + y) = f(x) + f(y) + 1 - \frac{2}{7}xy, \] then the value of \( 28 \sum_{i=1}^5 f(i) \) is:
Let \( f(x) = \frac{2^{x+2} + 16}{2^{2x+1} + 2^{x+4} + 32} \). Then the value of \[ 8 \left( f\left( \frac{1}{15} \right) + f\left( \frac{2}{15} \right) + \dots + f\left( \frac{59}{15} \right) \right) \] is equal to:
The area of the region enclosed by the curves \( y = e^x \), \( y = |e^x - 1| \), and the y-axis is:
The sum of all local minimum values of the function \( f(x) \) as defined below is:
\[ f(x) = \begin{cases} 1 - 2x & \text{if } x < -1, \\[10pt] \frac{1}{3}(7 + 2|x|) & \text{if } -1 \leq x \leq 2, \\[10pt] \frac{11}{18}(x-4)(x-5) & \text{if } x > 2. \end{cases} \]
If $ 0 \le x \le 3,\ 0 \le y \le 3 $, then the number of solutions $(x, y)$ for the equation: $$ \left( \sqrt{\sin^2 x - \sin x + \frac{1}{2}} \right)^{\sec^2 y} = 1 $$
Assertion (A): Let \( A = \{ x \in \mathbb{R} : -1 \leq x \leq 1 \} \). If \( f : A \to A \) be defined as \( f(x) = x^2 \), then \( f \) is not an onto function.
Reason (R): If \( y = -1 \in A \), then \( x = \pm \sqrt{-1} \notin A \).
In an arithmetic progression, if \( S_{40} = 1030 \) and \( S_{12} = 57 \), then \( S_{30} - S_{10} \) is equal to:
The area of the region enclosed by the curves \( y = e^x \), \( y = |e^x - 1| \), and the y-axis is:
Let \( f: [0, 3] \to A \) be defined by \( f(x) = 2x^3 - 15x^2 + 36x + 7 \) and \( g: [0, \infty) \to B \) be defined by \( g(x) = \frac{x}{x^{2025} + 1}. \) If both functions are onto and \( S = \{ x \in \mathbb{Z} : x \in A { or } x \in B \} \), then \( n(S) \) is equal to:
In \( I(m, n) = \int_0^1 x^{m-1} (1-x)^{n-1} \, dx \), where \( m, n > 0 \), then \( I(9, 14) + I(10, 13) \) is: