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\QTR{small}{Matematick\U{e1} anal\U{fd}za I online - Funkcie - Ot\U{e1}zky\dotfill \thepage }} %} \newtheorem{theorem}{Theorem} \newtheorem{acknowledgement}[theorem]{Acknowledgement} \newtheorem{algorithm}[theorem]{Algorithm} \newtheorem{axiom}[theorem]{Axiom} \newtheorem{case}[theorem]{Case} \newtheorem{claim}[theorem]{Claim} \newtheorem{conclusion}[theorem]{Conclusion} \newtheorem{condition}[theorem]{Condition} \newtheorem{conjecture}[theorem]{Conjecture} \newtheorem{corollary}[theorem]{Corollary} \newtheorem{criterion}[theorem]{Criterion} \newtheorem{definition}[theorem]{Definition} \newtheorem{example}[theorem]{Example} \newtheorem{exercise}[theorem]{Exercise} \newtheorem{lemma}[theorem]{Lemma} \newtheorem{notation}[theorem]{Notation} \newtheorem{problem}[theorem]{Problem} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{remark}[theorem]{Remark} \newtheorem{solution}[theorem]{Solution} \newtheorem{summary}[theorem]{Summary} \newenvironment{proof}[1][Proof]{\textbf{#1.} }{\ \rule{0.5em}{0.5em}} \input{tcilatex} \begin{document} \author{A. U. Thor} \title{Lab Report} \date{The Date } \maketitle \begin{abstract} A Laboratory report created with Scientific Notebook \end{abstract} \section{Funkcie} \begin{center} \begin{tabular}{|c|c|c|c|} \hline \textbf{% %TCIMACRO{\hyperref{Obsah}{}{}{maindex.tex}}% %BeginExpansion \msihyperref{Obsah}{}{}{maindex.tex}% %EndExpansion } & \textbf{% %TCIMACRO{\hyperref{Obsah kapitoly}{}{}{M2.tex}}% %BeginExpansion \msihyperref{Obsah kapitoly}{}{}{M2.tex}% %EndExpansion } & \textbf{% %TCIMACRO{\hyperref{Cvi\v{c}enia}{}{}{C2.tex}}% %BeginExpansion \msihyperref{Cvi\v{c}enia}{}{}{C2.tex}% %EndExpansion } & \textbf{% %TCIMACRO{\hyperref{Index}{}{}{G1.tex}}% %BeginExpansion \msihyperref{Index}{}{}{G1.tex}% %EndExpansion } \\ \hline \end{tabular} \end{center} \section{Ot\'{a}zky} Preverte si znalosti z\'{\i}skan\'{e} v tejto kapitole. Zodpovedajte v\v{s}% etky ot\'{a}zky. Ak neviete nejak\'{u} ot\'{a}zku zodpoveda\v{t} znovu pre% \v{s}tudujte pr\'{\i}slu\v{s}n\'{u} \v{c}as\v{t} a op\"{a}\v{t} odpovedajte. Po d\^{o}kladnom preveren\'{\i} Va\v{s}ich vedomost\'{\i} sa venujte po\v{c}% \'{\i}taniu pr\'{\i}kladov. \begin{itemize} \item Definujte funkciu. \item Kedy hovor\'{\i}me o re\'{a}lnej funkcii re\'{a}lnej premennej? \item Definujte zdola ohrani\v{c}en\'{u} funkciu na podmno\v{z}ine defini% \v{c}n\'{e}ho oboru a uve\v{d}te pr\'{\i}klad. \item Definujte zhora ohrani\v{c}en\'{u} funkciu na podmno\v{z}ine defini% \v{c}n\'{e}ho oboru a uve\v{d}te pr\'{\i}klad. \item Definujte infimum a supremum funkcie na podmno\v{z}ine defini\v{c}n% \'{e}ho oboru a uve\v{d}te pr\'{\i}klad. \item Definujte minimum a maximum funkcie a uve\v{d}te pr\'{\i}klad. \item Definujte funkciu rast\'{u}cu na podmno\v{z}ine defini\v{c}n\'{e}ho oboru a uve\v{d}te pr\'{\i}klad. \item Definujte funkciu klesaj\'{u}cu na podmno\v{z}ine defini\v{c}n\'{e}ho oboru a uve\v{d}te pr\'{\i}klad. \item Definujte funkciu neklesaj\'{u}cu na podmno\v{z}ine defini\v{c}n\'{e}% ho oboru a uve\v{d}te pr\'{\i}klad. \item Definujte funkciu nerast\'{u}cu na podmno\v{z}ine defini\v{c}n\'{e}ho oboru a uve\v{d}te pr\'{\i}klad. \item Definujte p\'{a}rnu funkciu a uve\v{d}te pr\'{\i}klad. \item Definujte nep\'{a}rnu funkciu a uve\v{d}te pr\'{\i}klad. \item Dan\'{a} je funkcia $f$\ , tak\'{a} \v{z}e $D\left( f\right) =\left\langle -3,8\right\rangle .$ M\^{o}\v{z}e by\v{t} funkcia $f$\ p\'{a}% rna alebo nep\'{a}rna? \item Definujte periodick\'{u} funkciu. \item Dan\'{a} je funkcia $f$, ktor\'{a} m\'{a} jeden nulov\'{y} bod ($% \exists !c\in D\left( f\right) :f\left( c\right) =0$). M\^{o}\v{z}e by\v{t} funkcia $f$ periodickou funkciou? \item Definujte graf funkcie. \item Je pravdiv\'{e} tvrdenie: \quotedblbase priamka rovnobe\v{z}n\'{a} s osou $o_{y}$ pret\'{\i}na graf funkcie v dvoch alebo viacer\'{y}ch bodoch''? Pre\v{c}o? \item Je pravdiv\'{e} tvrdenie: \quotedblbase funkcia $f$ je injekt\'{\i}vna vtedy a len vtedy ak priamka rovnobe\v{z}n\'{a} s osou $o_{x}$ pret\'{\i}na graf funkcie \ $f$ \ najviac v jednom bode\textquotedblright ? \item Defini\v{c}n\'{y}m oborom s\'{u}\v{c}tu dvoch funkci\'{\i} je zjednotenie ich defini\v{c}n\'{y}ch oborov? \item Defini\v{c}n\'{y}m oborom s\'{u}\v{c}inu dvoch funkci\'{\i} je kart% \'{e}zky s\'{u}\v{c}in ich defini\v{c}n\'{y}ch oborov? \item Definujte zlo\v{z}en\'{u} funkciu $f\circ g$. \item Ako n\'{a}jdeme defini\v{c}n\'{y} obor zlo\v{z}enej funkcie $f\circ g$? \item Je oper\'{a}cia skladania funkci\'{\i} komutat\'{\i}vna? Uve\v{d}te pr% \'{\i}klad. \item Za ak\'{y}ch podmienok existuje inverzn\'{a} funkcia k funkcii $% f:A\longrightarrow B,\;A,\,B\subset \mathbf{R?}$ \item Graf funkcie a k nej inverznej funkcie s\'{u} s\'{u}mern\'{e} pod\v{l}% a osi $o_{y}$? Uve\v{d}te pr\'{\i}klad. \item Funkcia $f:\mathbf{R}\longrightarrow \mathbf{R,\,}f\left( x\right) =x^{n},\,n\in \mathbf{N}$ \ je rast\'{u}ca? \item Funkcie s\'{\i}nus a kos\'{\i}nus s\'{u} periodick\'{e} funkcie? Ak% \'{u} maj\'{u} peri\'{o}du? \item Ak\'{y} je defini\v{c}n\'{y} obor mocninovej funkcie s prirodzen\'{y}m exponentom? \item Ak\'{a} je z\'{a}kladn\'{a} peri\'{o}da funkcie $\log _{a}x$? \item Pre ak\'{y} z\'{a}klad $a$ je funkcia $\log _{a}x$ rast\'{u}ca? \item Pre ak\'{y} z\'{a}klad $a$ je funkcia $a^{x}$ klesaj\'{u}ca? \item Uve\v{d}te pr\'{\i}klad funkcie, ktorej graf pretne priamka $y=k$ pre $% k\in \left( 0,\infty \right) $ dvakr\'{a}t. \end{itemize} \begin{center} \begin{tabular}{|c|c|c|c|} \hline \textbf{% %TCIMACRO{\hyperref{Obsah}{}{}{maindex.tex}}% %BeginExpansion \msihyperref{Obsah}{}{}{maindex.tex}% %EndExpansion } & \textbf{% %TCIMACRO{\hyperref{Obsah kapitoly}{}{}{M2.tex}}% %BeginExpansion \msihyperref{Obsah kapitoly}{}{}{M2.tex}% %EndExpansion } & \textbf{% %TCIMACRO{\hyperref{Cvi\v{c}enia}{}{}{C2.tex}}% %BeginExpansion \msihyperref{Cvi\v{c}enia}{}{}{C2.tex}% %EndExpansion } & \textbf{% %TCIMACRO{\hyperref{Index}{}{}{G1.tex}}% %BeginExpansion \msihyperref{Index}{}{}{G1.tex}% %EndExpansion } \\ \hline \end{tabular} \end{center} \rule{6.5in}{0.04in} \textsl{Matematick\'{a} anal\'{y}za I} \section{Funkcie} \end{document}