%% This document created by Scientific Notebook (R) Version 3.5 %% Starting shell: article \documentclass{article} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \usepackage{amssymb} %TCIDATA{OutputFilter=LATEX.DLL} %TCIDATA{Version=5.00.0.2570} %TCIDATA{} %TCIDATA{Created=Wednesday, February 10, 1999 13:29:48} %TCIDATA{LastRevised=Sunday, February 13, 2005 15:34:41} %TCIDATA{} %TCIDATA{} %TCIDATA{CSTFile=On line bluem.cst} %TCIDATA{PageSetup=72,72,72,72,0} %TCIDATA{Counters=arabic,1} %TCIDATA{AllPages= %H=36 %F=36,\PARA{038
\QTR{small}{Matematick\U{e1} anal\U{fd}za I online - Funkcie - Element\U{e1}rne funkcie\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|c|c|} \hline \textbf{% %TCIMACRO{% %\hyperref{\hyperref{\hyperref{Obsah}{}{}{maindex.tex}}{}{}{M2.tex}}{}{}{maindex.tex}}% %BeginExpansion \msihyperref{% \msihyperref{% \msihyperref{Obsah}{}{}{maindex.tex}}{}{}{M2.tex}}{}{}{maindex.tex}% %EndExpansion } & \textbf{% %TCIMACRO{\hyperref{Obsah kapitoly}{}{}{M2.tex}}% %BeginExpansion \msihyperref{Obsah kapitoly}{}{}{M2.tex}% %EndExpansion } & \textbf{% %TCIMACRO{\hyperref{Predch\'{a}dzaj\'{u}ca str\'{a}nka}{}{}{M25.tex}}% %BeginExpansion \msihyperref{Predch\'{a}dzaj\'{u}ca str\'{a}nka}{}{}{M25.tex}% %EndExpansion } & \textbf{% %TCIMACRO{\hyperref{Ot\'{a}zky}{}{}{O2.tex}}% %BeginExpansion \msihyperref{Ot\'{a}zky}{}{}{O2.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} \subsection{Element\'{a}rne funkcie\label{11}} V tejto \v{c}asti zopakujeme z\'{a}kladn\'{e} vlastnosti element\'{a}rnych funkci\'{\i}, ktor\'{e} budeme pou\v{z}\'{\i}va\v{t} v \v{d}al\v{s}\'{\i}ch kapitol\'{a}ch. \begin{definition} \emph{Kon\v{s}tantn\'{a} funkcia }$f:\mathbf{R}\longrightarrow \left\{ c\right\} \mathbf{,\ }f\left( x\right) =c\in \mathbf{R}$. Grafom kon\v{s}% tantnej funkcie je priamka rovnobe\v{z}n\'{a} s osou $o_{x}$. \label{2}\emph{Mocninov\'{a} funkcia s prirodzen\'{y}m exponentom} $n\in \mathbf{N}$ je funkcia \[ f:\mathbf{R}\longrightarrow \mathbf{R,\ }f\left( x\right) =x^{n}. \] \label{3}\emph{Mocninov\'{a} funkcia so z\'{a}porn\'{y}m cel\'{y}m exponentom% } $-n$, kde $n\in \mathbf{N}$ je funkcia \[ f:\left( -\infty ,0\right) \cup \left( 0,\infty \right) \longrightarrow \mathbf{R,\ }f\left( x\right) =x^{-n}=\frac{1}{x^{n}}. \] \label{4}Funkcia \emph{n-t\'{a} odmocnina }($n\in \mathbf{N,\;}n\geq 2$) je definovan\'{a} \[ f:\left\{ \begin{tabular}{ccc} $\left\langle 0,\infty \right) \longrightarrow \left\langle 0,\infty \right) \mathbf{,\ }f\left( x\right) =\sqrt[n]{x}=x^{\frac{1}{n}},$ & pre & $n$ p% \'{a}rne \\ $\mathbf{R}\longrightarrow \mathbf{R,\ }f\left( x\right) =\sqrt[n]{x}=x^{% \frac{1}{n}},$ & pre & $n$ nep\'{a}rne,% \end{tabular}% \right. \]% je rast\'{u}ca funkcia. \label{5}\emph{Algebrick\'{y} polyn\'{o}m} \[ P:\mathbf{R}\longrightarrow \mathbf{R,\ }P\left( x\right) =c_{0}x^{n}+c_{1}x^{n-1}+c_{2}x^{n-2}+...+c_{n-1}x+c_{n},c_{0}\neq 0, \]% kde$\;c_{i}\in \mathbf{R,}\;i=0,1,...,n$ naz\'{y}vame koeficienty polyn\'{o}% mu $f$ a $n\in \mathbf{N}$, stupe\v{n} polyn\'{o}mu $P$. \label{6}\emph{Racion\'{a}lna funkcia }% \[ R:\mathbf{R\setminus }Q^{\circ }\longrightarrow \mathbf{R,\ }R\left( x\right) =\frac{P\left( x\right) }{Q\left( x\right) }, \] kde $P,Q$ s\'{u} polyn\'{o}my a $Q^{\circ }=\left\{ x\in \mathbf{R}:Q\left( x\right) =0\right\} $. \label{7}\emph{Exponenci\'{a}lna funkcia }% \[ f:\mathbf{R}\longrightarrow \left( 0,\infty \right) \mathbf{,\ }f\left( x\right) =a^{x},\;\left( a>0,\;a\neq 1\right) . \] Pre $a>1$ je $f\left( x\right) =a^{x}$ rast\'{u}ca funkcia, pre $01,$ klesaj\'{u}ca pre $01$, klesaj\'{u}ca pre $0