%% 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 15:33:52} %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 - Inverzn\U{e9} 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|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{Predch\'{a}dzaj\'{u}ca str\'{a}nka}{}{}{M24.tex}}% %BeginExpansion \msihyperref{Predch\'{a}dzaj\'{u}ca str\'{a}nka}{}{}{M24.tex}% %EndExpansion } & \textbf{% %TCIMACRO{\hyperref{Nasleduj\'{u}ca str\'{a}nka}{}{}{M26.tex}}% %BeginExpansion \msihyperref{Nasleduj\'{u}ca str\'{a}nka}{}{}{M26.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{Inverzn\'{e} funkcie} Nech $f:A\longrightarrow B$, je funkcia (zobrazenie) definovan\'{a} na mno% \v{z}ine $A$\ s hodnotami v mno\v{z}ine $B.$ Hovor\'{\i}me, \v{z}e $f$\ je zobrazenie mno\v{z}iny $A$ \emph{do }mno\v{z}iny $B.$\ Nech $M\subset A.$ Ozna\v{c}me \[ f\left( M\right) =\left\{ y\in B\,;\,\left( y=f\left( x\right) \right) \wedge \left( x\in M\right) \right\} . \] \begin{definition} Nech $f:A\longrightarrow B$, je funkcia (zobrazenie) definovan\'{e} na mno% \v{z}ine $A$\ s hodnotami v mno\v{z}ine $B.$ Ak $f\left( A\right) =B,$ hovor% \'{\i}me, \v{z}e $f$ je zobrazen\'{\i}m \emph{na} mno\v{z}inu $B$ , alebo $f$ je \emph{surjekt\'{\i}vna funkcia (surjekcia)}. \end{definition} \begin{definition} Funkciu (zobrazenie) $f:A\longrightarrow B$ naz\'{y}vame \emph{injekt\'{\i}% vnou (prostou)} ak $\forall x_{1},x_{2}\in A\,;\,x_{1}\neq x_{2}$ plat\'{\i} $f(x_{1})\neq f(x_{2})$. \end{definition} \begin{definition} Funkciu (zobrazenie) $f:A\longrightarrow B,$ ktor\'{a} je injekt\'{\i}vna a surjekt\'{\i}vna naz\'{y}vame \emph{bijekt\'{\i}vna funkcia (bijekcia).} \end{definition} \begin{definition} \label{1}Nech $f:A\longrightarrow B$, je bijekcia. Funkciu $% f^{-1}:B\longrightarrow A$, definovan\'{u} tak, \v{z}e $f^{-1}\left( y\right) =x$ pr\'{a}ve vtedy, ke\v{d} $f(x)=y$, naz\'{y}vame \emph{inverznou funkciou} \emph{k funkcii} $f:A\longrightarrow B$. \end{definition} \begin{example} Nech $h:\left\langle 0,\infty \right) \longrightarrow \left\langle 0,\infty \right) ,\,h(x)=x^{2}$. N\'{a}jdite inverzn\'{u} funkciu $h^{-1}$. \end{example} \begin{solution} M\'{a}me $A=D\left( h\right) =B$. Plat\'{\i} \[ \forall x_{1},x_{2}\in \left\langle 0,\infty \right) \,;\,x_{1}\neq x_{2}\Longrightarrow f(x_{1})=x_{1}^{2}\neq x_{2}^{2}=f(x_{2}), \]% teda funkcia $h$ je injekcia. Nech $y\in \left\langle 0,\infty \right) ,$ potom $\exists !\ x\in \left\langle 0,\infty \right) $ tak\'{e}, \v{z}e $% y=x^{2}$ ($x=\sqrt{y}$), teda funkcia $h$ je surjekcia. Preto $% h:\left\langle 0,\infty \right) \longrightarrow \left\langle 0,\infty \right) ,\,h(x)=x^{2}$ je bijekcia a existuje inverzn\'{a} funkcia $% h^{-1}:\left\langle 0,\infty \right) \longrightarrow \left\langle 0,\infty \right) ,\,h^{-1}(t)=\sqrt{t}$. Na obr\'{a}zku vid\'{\i}me grafy oboch funkci% \'{\i}:$x^{2}$\FRAME{dtbpFX}{4.4996in}{3in}{0pt}{}{}{Plot}{\special{language "Scientific Word";type "MAPLEPLOT";width 4.4996in;height 3in;depth 0pt;display "USEDEF";plot_snapshots TRUE;mustRecompute FALSE;lastEngine "MuPAD";xmin "0";xmax "2";xviewmin "-0.002";xviewmax "2.002";yviewmin "-0.004";yviewmax "4.004";plottype 4;constrained TRUE;numpoints 100;plotstyle "patch";axesstyle "normal";xis \TEXUX{x};yis \TEXUX{y};var1name \TEXUX{$x$};var2name \TEXUX{$y$};function \TEXUX{$x^{2}$};linecolor "green";linestyle 1;pointstyle "point";linethickness 2;lineAttributes "Solid";var1range "0,2";num-x-gridlines 100;curveColor "[flat::RGB:0x0000ff00]";curveStyle "Line";rangeset"X";function \TEXUX{$\sqrt{x}$};linecolor "magenta";linestyle 1;pointstyle "point";linethickness 2;lineAttributes "Solid";var1range "0,2";num-x-gridlines 100;curveColor "[flat::RGB:0x00ff00ff]";curveStyle "Line";valid_file "T";tempfilename 'HTRF5K06.wmf';tempfile-properties "XPR";}% } $\square $ \end{solution} \begin{example} Nech $f:\mathbf{R}\longrightarrow \mathbf{R,\,}f(x)=2x+1$. N\'{a}jdite inverzn\'{u} funkciu $f^{-1}$. \end{example} \begin{solution} Preto\v{z}e $f$ je bijekcia, inverzn\'{a} funkcia existuje a m\'{a}me: $% f^{-1}:\mathbf{R}\longrightarrow \mathbf{R,\,}f^{-1}(x)=\frac{x-1}{2}.$ Na obr\'{a}zku vid\'{\i}me grafy oboch funkci\'{\i}:$2x+1$\FRAME{dtbpFX}{% 4.4996in}{3in}{0pt}{}{}{Plot}{\special{language "Scientific Word";type "MAPLEPLOT";width 4.4996in;height 3in;depth 0pt;display "USEDEF";plot_snapshots TRUE;mustRecompute FALSE;lastEngine "MuPAD";xmin "-5";xmax "5";xviewmin "-5.01";xviewmax "5.01";yviewmin "-9.02";yviewmax "11.02";plottype 4;constrained TRUE;numpoints 100;plotstyle "patch";axesstyle "normal";xis \TEXUX{x};yis \TEXUX{y};var1name \TEXUX{$x$};var2name \TEXUX{$y$};function \TEXUX{$2x+1$};linecolor "green";linestyle 1;pointstyle "point";linethickness 2;lineAttributes "Solid";var1range "-5,5";num-x-gridlines 100;curveColor "[flat::RGB:0x0000ff00]";curveStyle "Line";function \TEXUX{$\frac{x-1}{2}$};linecolor "magenta";linestyle 1;pointstyle "point";linethickness 2;lineAttributes "Solid";var1range "-5,5";num-x-gridlines 100;curveColor "[flat::RGB:0x00ff00ff]";curveStyle "Line";valid_file "T";tempfilename 'HTRF9U07.wmf';tempfile-properties "XPR";}% } $\square $ \end{solution} \begin{example} Nech $f:\mathbf{R\setminus }\left\{ \frac{1}{2}\right\} \longrightarrow \mathbf{R\setminus }\left\{ 2\right\} \mathbf{,}$ $f(x)=\frac{4x+3}{2x-1}$. N% \'{a}jdite inverzn\'{u} funkciu $f^{-1}$. \end{example} \begin{solution} Funkcia $f$ je prost\'{a} (injekt\'{\i}vna): $\forall x_{1},x_{2}\in \mathbf{% R\setminus }\left\{ \frac{1}{2}\right\} :x_{1}\neq x_{2}\Rightarrow \frac{% 4x_{1}+3}{2x_{1}-1}\neq \frac{4x_{2}+3}{2x_{2}-1}$ t.j. $f(x_{1})\neq f(x_{2})$.\ Tvrdenie uk\'{a}\v{z}eme sporom. Ak by $f(x_{1})=f(x_{2})% \Rightarrow \frac{4x_{1}+3}{2x_{1}-1}=\frac{4x_{2}+3}{2x_{2}-1}\Rightarrow x_{1}=x_{2}$, pre $x_{1},x_{2}\neq \frac{1}{2}.$ Nech $y\in \mathbf{% R\setminus }\left\{ 2\right\} $, h\v{l}ad\'{a}me nejak\'{e} $x\in \mathbf{% R\setminus }\left\{ \frac{1}{2}\right\} $ aby $y=f(x)$. Teda $y=\frac{4x+3}{% 2x-1}\Longrightarrow x=\frac{y+3}{2y-4}$, ak $y\neq 2$. Teda $f$ je surjekcia. Preto\v{z}e $f$ je bijekcia, existuje inverzn\'{a} funkcia $% f^{-1}:\mathbf{R\setminus }\left\{ 2\right\} \longrightarrow \mathbf{% R\setminus }\left\{ \frac{1}{2}\right\} ,$ $f^{-1}(y)=\frac{y+3}{2y-4}$. Na% \v{c}rtnite graf funkcie $f$\ aj $f^{-1}.$ $\square $ \end{solution} \begin{description} \item[Pozn\'{a}mka] Nech $f:A\longrightarrow B$ je bijekcia. Potom z defin% \'{\i}cie inverznej funkcie $f^{-1}:B\longrightarrow A$ vypl\'{y}va, \v{z}e $% f(x)=y\Longleftrightarrow f^{-1}\left( y\right) =x$. Odtia\v{l} dostaneme \item $\forall x\in A$ je $\left( f^{-1}\circ f\right) \left( x\right) =x$, \item $\forall y\in B$ je $\left( f\circ f^{-1}\right) \left( y\right) =y$, to znamen\'{a} \item $f^{-1}\circ f:A\longrightarrow A,\ \left( f^{-1}\circ f\right) \left( x\right) =x$, \item $f\circ f^{-1}:B\longrightarrow B,\ \left( f\circ f^{-1}\right) \left( y\right) =y$. \end{description} Ak $f:A\longrightarrow B$ je bijekcia a $f^{-1}:B\longrightarrow A$ je k nej inverzn\'{a} funkcia, potom \begin{center} $G(f)=\left\{ \left( x,f\left( x\right) \right) \in A\times B\,;\,x\in A\right\} $ $G(f^{-1})=\left\{ \left( y,\,f^{-1}\left( y\right) \right) =\left( f\left( x\right) ,x\right) \in B\times A\,;\,y\in B\right\} .$ \end{center} \begin{quote} Teda graf funkcie $f$ a graf funkcie $f^{-1}$ s\'{u} s\'{u}mern\'{e} pod\v{l}% a priamky $y=x$. \end{quote} \begin{center} \begin{tabular}{|c|c|c|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{Predch\'{a}dzaj\'{u}ca str\'{a}nka}{}{}{M24.tex}}% %BeginExpansion \msihyperref{Predch\'{a}dzaj\'{u}ca str\'{a}nka}{}{}{M24.tex}% %EndExpansion } & \textbf{% %TCIMACRO{\hyperref{Nasleduj\'{u}ca str\'{a}nka}{}{}{M26.tex}}% %BeginExpansion \msihyperref{Nasleduj\'{u}ca str\'{a}nka}{}{}{M26.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} \rule{6.5in}{0.04in} \textsl{Matematick\'{a} anal\'{y}za I} \section{Funkcie} \end{document}