Functions

Among the following equations the one that most accurately describes the graph shown below is

The number of solutions of the equation $|\sin(\pi x)| = \dfrac{1}{50}(x^2 + 1)$ in $\mathbb{R}$ is

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

Let $A$ and $B$ be two sets. Let $A_1, A_2$ and $A_3$ be subsets of $A$ such that $A_1 \cup A_2 \cup A_3=A$ and $A_i \cap A_j=\phi$ for $i \neq j$. Let $f: A \rightarrow B$ be a function. For each $i=1,2,3$, let $f_i:…

For $x \in \mathbb{R}$, define $\langle x\rangle=x-[x]$. Then for any arbitrary pair of real numbers $x, y \in \mathbb{R}^{+}$,

For $m, n \in \mathbb{Z}$, let $f_{m, n}: \mathbb{Z} \rightarrow \mathbb{Z}$ be the function defined as $f_{m, n}(x)=m x+n \quad$ for all $x \in \mathbb{Z}$. Let $\mathcal{F}$ be the collection of all such functions,…

Let $f: \mathbb{Z} \rightarrow \mathbb{R}$ be a function such that $f(m+n) f(m) f(n)=1$ for all $m, n$ in $\mathbb{Z}$. Then

Let $S$ be the set of all functions $f: \mathbb{Z} \rightarrow \mathbb{C}$ such that $f(m+n)=f(m) f(n)$ for all $m, n \in \mathbb{Z}$ and $f(0)=1$. Then

Let $f:[0,1] \rightarrow \mathbb{R}$ be a continuous function and $P$ be a polynomial of degree 4 with coefficients in $\mathbb{R}$. If $P(f(x))=0$ for all $x \in \mathbb{R}$, then

Let $X, Y, Z$ be sets and $f: X \rightarrow Y$ and $g: Y \rightarrow Z$ be functions. Then

Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a function defined by $f(x)=|x|+2 x$ for $x \in \mathbb{R}$. Then $f$ is

Let $f:[0,1] \rightarrow[0,1]$ be a non-constant continuous function different from the identity function. Then