Consider two positive real numbers $ a,b $ and the function $ f:(0,\infty )\longrightarrow\left( \sqrt{ab} ,\frac{a+b}{2} \right) $ defined as $ f(x)=-x+\sqrt{x^2+(a+b)x+ab}. $ Prove that it's bijective. [i]D.M. Bătineți-Giurgiu[/i] and [i]Neculai Stanciu[/i]

Let $ M $ be a set of nonzero real numbers and $ f:M\longrightarrow M $ be a function having the property that the identity function is $ f+f^{-1} . $ [b]1)[/b] Prove that $ m\in M\iff -m\in M. $ [b]2)[/b] Show that $ f $ is odd. [b]3)[/b] Determine the cardinal of $ M. $

Consider a countinuous function $ f:\mathbb{R}_{>0}\longrightarrow\mathbb{R}_{>0} $ that verifies the following conditions: $ \text{(1)} x f(f(x))=(f(x))^2,\quad\forall x\in\mathbb{R}_{>0} $ $ \text{(2)} \lim_{\stackrel{x\to 0}{x>0}} \frac{f(x)}{x}\in\mathbb{R}\cup\{ \pm\infty \} $ [b]a)[/b] Show that $ f $ is bijective. [b]b)[/b] Prove that the sequences $ \left( (\underbrace{f\circ f\circ\cdots \circ f}_{\text{n times}} ) (x) \right)_{n\ge 1} ,\left( (\underbrace{f^{-1}\circ f^{-1}\circ\cdots \circ f^{-1}}_{\text{n times}} ) (x) \right)_{n\ge 1} $ are both arithmetic progressions, for any fixed $ x\in\mathbb{R}_{>0} . $ [b]c)[/b] Determine the function $ f. $ [i]Nelu Chichirim[/i]

[b]a)[/b] Prove that the function $ f:\mathbb{Z}_{\ge 0}\longrightarrow\mathbb{Z}_{\ge 0} , $ given as $ f(n)=n+(-1)^n $ is bijective. [b]b)[/b] Find all surjective functions $ g:\mathbb{Z}_{\ge 0}\longrightarrow\mathbb{Z}_{\ge 0} $ that have the property that $ g(n)\ge n+(-1)^n , $ for any nonnegative integer.

Consider the parametric function $ f_k:\mathbb{R}\longrightarrow\mathbb{R}, f(x)=x+k\lfloor x \rfloor . $ [b]a)[/b] For which integer values of $ k $ the above function is injective? [b]b)[/b] For which integer values of $ k $ the above function is surjective? [b]c)[/b] Given two natural numbers $ n,m, $ create two bijective functions: $$ \phi : f_m (\mathbb{R} )\cap [0,\infty )\longrightarrow f_n(\mathbb{R})\cap [0,\infty ) $$ $$ \psi : \left(\mathbb{R}\setminus f_m (\mathbb{R})\right)\cap [0,\infty )\longrightarrow\left(\mathbb{R}\setminus f_n (\mathbb{R})\right)\cap [0,\infty ) $$ [i]Cristinel Mortici[/i]

Let be three finite and nonempty sets $ A,B,C $ such that $ |A|=|C|\le |B| , $ and a bijection $ A\stackrel{\beta }{\longrightarrow } C. $ How many pairs of functions $ A\stackrel{f_2 }{\longrightarrow } B\stackrel{f_1 }{\longrightarrow } C $ that satisfy $ f_1\circ f_2=\beta $ are there?

Find all functions $f:\mathbb {R}^{+} \rightarrow \mathbb{R}^{+}$, such that $$f(af(b)+a)(f(bf(a))+a)=1$$ for any positive reals $a, b$.