\subsubsection {}
Nech� $M\subset\Bbb R$ je aspo�
dvouprvkov�,
$f:M\to\Bbb R$. Plat� v�dy :

a) $f$ je nerostouc� na $M$ $\Rightarrow$ $f$ nen�
rostouc� na
$M$ ?

b) $f$  nen� rostouc� na $M$ $\Rightarrow$ $f$ je
nerostouc� na
$M$ ?

c) ($f$  nen� rostouc� na ��dn� dvouprvkov�
$A\subset M$)
$\Rightarrow$
$f$ je nerostouc� na $M$ ?



\subsubsection {}
Nech� $f_1,f_2:(0,1)\to\Bbb R$ jsou
rostouc�
funkce a $g_1,g_2:\Bbb R\to(0,1)$ jsou klesaj�c�.
Kter� z
n�sleduj�c�ch funkc� jsou v�dy
monot�nn� ?

(i) $g_1\circ f_2\circ g_2\circ f_1$,\quad (ii)
$\max(f_1,f_2)\circ g_1\circ (f_1+f_2)$,\quad (iii)
$(f_1f_2)\circ g_1$,

(iv) $(g_1g_2)\circ\min(f_1,f_2)$.



\subsubsection {}
P�eformulujte co nejjednodu��eji
($a,A\in\Bbb
R$):

(i) $(\forall \varepsilon >0)(\exists K>0)(\exists \delta>0)(\forall
x)(x\in P_\delta(a)\Rightarrow\vert f(x)-A\vert < K \varepsilon)$,\quad

(ii) $(\exists\delta>0)(\forall \varepsilon >0)(\forall x\in\Bbb R)
 (x\in P_\delta(a)\Rightarrow\vert f(x)-A\vert < \varepsilon )$.


\subsubsection {}
Najd�te re�lnou ��st ��sla

$1+2(\cos\frac\pi3 +
i\sin\frac\pi3)+4(\cos\frac{2\pi}3+i\sin\frac{2\pi}3)+\dots+ 2^{100}
(\cos\frac{100\pi}3+i\sin\frac{100\pi}3).$

\subsubsection {}
Nech� $f(x)={2\sqrt x\over 4-\sqrt x}$. Nalezn�te $D_f,\,
H_f,\, f^{-1}$. 

\subsubsection {}
Nech� $\varphi:<0,\infty)\to<1,\infty)$ je bijekce a
$\psi(x)=\sqrt{\varphi^2(x)-1}$. Doka�te, �e existuje inverzn� funkce
$\psi^{-1}$, a vyj�d�ete ji pomoc� $\varphi^{-1}$. Co je
$D_{\psi^{-1}}$? 

\subsubsection {}
 Nech� $f:\Bbb C\to\Bbb C,\,g:\Bbb C\to\Bbb C,\, f(z)=z+2\bar
z,\, g(z)=a(2\bar z-z)$. Existuje $a\in\Bbb C$, pro kter� $g=f^{-1}$ ?


\subsubsection {}
Plat� v�rok
$(\exists M\subset\Bbb R)(\exists! f:M\to\Bbb R)(\forall x\in\Bbb
R)(x\ne 0\Rightarrow f(x+\frac1x)=g(x)),$

je-li
$a)\,g(x)=\sin x+\sin\frac1x,\quad b)\,g(x)=\sqrt{x^2+1},\quad c)\,g(x)=
{x^{16}+1\over x^8}$ ?

Pokuste se charakterizovat v�echny funkce $g$, pro kter� uveden� v�rok
plat�.

\subsubsection {}
$\matrix \mbox{Plat� v�roky} &
(1)\, (\exists M\subset\Bbb R)(\exists! f:M\to\Bbb R)(\forall x\in\Bbb
R)(f(x^2-2x)=g(x)),        \\
&(2)\, (\exists M\subset\Bbb R)(\exists! f:M\to\Bbb R)(\forall x\in\Bbb
R)(f(x-\vert x\vert)=g(x)),\endmatrix$


je-li
$a)\,g(x)=\cos(1-x),\quad b)\,g(x)=\sin(1-x)\quad c)\,g(x)=\max(x,0),$

$d)\,g(x)= \max(-x,0)$ ?

Pokuste se charakterizovat v�echny funkce $g$, pro kter� uveden� v�roky
plat�.


