Recent questions tagged functions

Let $S$ be the set of nonnegative continuous functions $f$ on $[0,1]$ satisfying\[ \int_{0}^{1} \sin ^{2}(x) f(x) d x=\int_{0}^{1} \sin (x) \cos (x) f(x) d x=\int_{0}^{1}...

Which of the following is true for every function $u: \mathbb{R} \rightarrow \mathbb{R}$ which is continuously differentiable on $\mathbb{R} \;\text{(i.e}., u$ is difffer...

Please list out the best free available video playlist for Functions from Quantitative Aptitude as an answer here (only one playlist per answer). We'll then select the be...

Show that for every $\theta \in\left(0, \frac{\pi}{2}\right),$ there exists a unique real number $x_{\theta}$ such that $$ (\sin \theta)^{x_{\theta}}+(\cos \theta)^{x_{\t...

Consider the function $f: \mathbb{R}^{2} \rightarrow \mathbb{R}$ defined by $$ f(0,0)=0, \quad f(x, y)=\frac{x y}{x^{2}+y^{2}}, \quad(x, y) \neq(0,0) . $$ Prove that the ...

Let $f:[0,1] \rightarrow[0, \infty)$ be a continuous function. Let $$ a=\inf _{0 \leq x \leq 1} f(x) \text { and } b=\sup _{0 \leq x \leq 1} f(x) . $$ For every positive ...

Let $f_{1}:[0,4] \rightarrow[0,4]$ be defined by $f_{1}(x)=3-(x / 2)$. Define $f_{n}(x)=$ $f_{1}\left(f_{n-1}(x)\right)$ for $n \geq 2$.Prove that $\displaystyle{}\lim _{...

An $n$-variable Boolean function $f:\{0,1\}^{n} \rightarrow\{0,1\}$ is called symmetric if its value depends only on the number of $1 \text{'s}$ in the input. Let $\sigma...

Prove that the function defined by $$ f(x)=\sum_{n=0}^{\infty}\left(\frac{x^{n}}{n !}\right)^{2} $$ is continuous on $\mathbb{R}$, for any real number $x$.

Let $C_{0}$ be the set of all continuous functions $f:[0,1] \rightarrow \mathbb{R}$ and $C_{1}$ be the set of all differentiable functions $g:[0,1] \rightarrow \mathbb{R}...

Consider the function $f: \mathbb{C} \rightarrow \mathbb{C}$ defined on the complex plane $\mathbb{C}$ by $f(z)=e^{z}$. For a real number $c>0$, let $A=\{f(z) \mid \opera...

Consider two real valued functions $f$ and $g$ given by $$f(x)=\frac{x}{x-1} \; \text{for } x>1, \quad \text{and }\quad g(x)=7-x^{3} \; \text{for } x \in \mathbb{R}.$$ Wh...