List of top Mathematics Questions on Functions

If $A$ and $B$ are the domain and range of the real valued function, \[ f(x)=\dfrac{|x|}{\sqrt{1-|x|}} \] then $A \cup B =\ ?$

Let $S_1$ and $S_2$ be two circles drawn inside a unit square ABCD, touching each other externally. Further, the circle $S_1$ touches the sides AD and DC; and the circle $S_2$ touches the sides AB and BC. If the area of $S_2$ is twice the area of $S_1$, then the radius of $S_1$ is

Let $f : \mathbb{R} \to \mathbb{R}$ be a function defined by $f(x) = x^5 + x^3$ and let $g(x) = f^{-1}(x)$ be the inverse of $f$. If $g''(-2) = \frac{a}{b}$ where $a$ and $b$ are positive coprime integers, then the value of $a$ is

Let $f : \mathbb{R} \to \mathbb{R}$ be a function defined by \[ f(x) = \begin{cases} x \left( \frac{e^{1/x} - e^{-1/x}}{e^{1/x} + e^{-1/x}} \right) & \text{if } x \neq 0 \\ 0 & \text{if } x = 0 \end{cases} \] Then

Let a variable line $L$ meet $x$-axis and $y$-axis at points A and B, respectively. Suppose the distance of the line $L$ from the origin is 3 units. Then the equation of the locus of the point C that divides the line segment AB internally in the ratio $2 : 1$ is

For a complex number $z = x + iy$, where $x, y \in \mathbb{R}$, denote $\hat{z} = y + ix$. The locus of $z$ satisfying $|z + \hat{z}| = |z - \hat{z}|$ in the complex plane is

If $x \in (0, 1)$ and $\sin^{-1} x - \sin^{-1} \frac{1}{4} = \frac{\pi}{3}$, then $x$ equals

The parabola $y = -x^2 + 16$ intersects the $x$-axis at points A and B. Further, the parabola intersects the line $y = 7$ at points M and N. Then the area of the quadrilateral with vertices A, B, M and N is

Let $f : \mathbb{R} \to \mathbb{R}$ be a twice differentiable function such that $f''(x) = -f(x)$. Let $h : \mathbb{R} \to \mathbb{R}$ be a differentiable function such that $h'(x) = f(x)^2 + (f'(x))^2$ for all $x \in \mathbb{R}$. If $h(0) = 1$ and $h\left(\frac{1}{3}\right) = \frac{5}{3}$, then the value of $h(100)$ is

Suppose $\lim_{x \to -2} \frac{bx^2 + 15x + 15 + b}{x^2 + x - 2} = L$, where $b$ and $L$ are real numbers. Then

The domain of the real function $f(x) = \sin^{-1}\left(\log_2\left(\frac{x^2}{2}\right)\right)$ is

If the domain and the range of the real valued function \[ f(x)=\frac{1}{\sqrt{|x|-[x]}} \] are A and B, then $A\cap B=$ ($R^{+}$ is set of positive real numbers and $Z^{+}$ is set of positive integers)

If \[ f(x)= \begin{cases} \dfrac{x-2}{|x-2|}+a, & x<2, \\[6pt] a+b, & x=2, \\[6pt] \dfrac{x-2}{|x-2|}+b, & x>2, \end{cases} \] is continuous at $x=2$, then

Mr. X has two investment options: A (8% p.a. compounded semi-annually) or B (7.6% p.a. compounded quarterly). Which of the following are TRUE?
(A) Effective rate for option A is 8.16%
(B) Effective rate for option B is 7.82%
(C) Effective rate for option B is 8.82%
(D) Option A is better than option B as an investment.

Let $\mathbb{N}, \mathbb{Z}$ and $\mathbb{R}$ be the set of natural numbers, integers and real numbers respectively, $[\cdot]$ denotes the greatest integer function. Match List-I with List-II:}

For $x \in \mathbb{R}$, define \[ \langle x \rangle = x-[x] \] where $[x]$ denotes the greatest integer less than or equal to $x$. Then for any arbitrary pair of real numbers $x,y \in \mathbb{R}$, which of the following is always true?

If \[ f(x)= \begin{cases} 1,& |x|\le 1\\ 0,& |x|>1 \end{cases} \] \[ g(x)= \begin{cases} 2-x^2,& |x|\le 2\\ 2,& |x|>2 \end{cases} \] and $h(x)=f(g(x))$, then the interval in which $h(x)=1$ for all values of $x$ is:

If \[ f(x)=|x-2|+|x+1|, \] then the minimum value of $f(x)$ is:

The function \[ f(x)=\sin\left(\log\left(x+\sqrt{x^2+1}\right)\right) \] is

The domain of the real-valued function \[ f(x)=\cos^{-1}\left(\frac{2-x}{4}\right)+[\log(3-x)]^{-1} \] is:

The domain of the real valued function $ f(x)=\log_{x-1}(3x+1) $ is

If $ f:A\rightarrow B $, $ g:B\rightarrow C $ are two functions such that $ gof: A\rightarrow C $ is an onto function, then it is necessary that