# Exponential Func

```1. Hatv&aacute;nyf&uuml;ggv&eacute;nyek
f(x)=xa (aR)
1.1. a = nN+
a)
b)
1.2. a = 1/n
a)
b)
1
1.3. a&lt;0 , aZ
a)
b)
2
1.4. &Ouml;sszehasonl&iacute;t&aacute;s: aR, 0&lt;x
3
2. Exponenci&aacute;lis &eacute;s logaritmus f&uuml;gv&eacute;nyek
2.1.
a)
b)
2.2.
a)
b)
4
2.3. &Ouml;sszefoglal&oacute; &aacute;br&aacute;k
5
3.a) Trigonometrikus f&uuml;ggv&eacute;nyek
6
3.b) Trigonometrikus f&uuml;ggv&eacute;nyek inverzei (arcus- fv.)
7
4. Hiperbolikus f&uuml;ggv&eacute;nyek &eacute;s inverzeik
e x  ex
(sinus hyperbolicus)
2
Dom(sh)=R, Im(sh)=R,
4.1. sh(x):=
p&aacute;ratlan fv., szig. mon. nő
=&gt;
sh-1(y) = Arsh(y) = ln( y 
lim sh(x) = -∞ , lim sh(x) = +∞ .

invert&aacute;lhat&oacute;: (Area sinus hyp.)

y 2  1)
e x  ex
(cosinus hyperbolicus)
2
Dom(ch)=R, Im(ch)=R,
4.2. ch(x):=
p&aacute;ros fv., x&gt;0 eset&eacute;n szig. mon. nő
lim ch(x) = lim ch(x) = +∞ .


=&gt;
invert&aacute;lhat&oacute;: (Area cosinus hyp.)
ch-1(y) = Arch(y) = ln( y 
y 2  1)
8
sh( x ) e x  e  x
=
ch( x ) e x  e  x
Dom(th)=R, Im(th)=(0,1),
4.3. th(x):=
(tangens hyperbolicus)
p&aacute;ratlan fv., szig. mon. nő
lim th(x) = -1 ,

4.4. cth(x):=
=&gt;
invert&aacute;lhat&oacute;: (Area tangens hyp.)
1 1 y 
th-1(y) = Arth(y) = ln 
 (|y|&lt;1)
2  1  y 
=&gt;
invert&aacute;lhat&oacute;: (Area cotangens hyp.)
1 1 y 
cth-1(y) = Arcth(y)= ln 
 (|y|&gt;1)
2  1  y 
lim th(x) = +1 .

e x  ex
1
= x
(cotangens
th( x ) e  e  x
hyperbolicus)
Dom(cth)=R\{0}, Im(cth)=R\[0,1],
p&aacute;ratlan fv., k&eacute;t &aacute;ga szig. mon. cs&ouml;kken
lim cth(x) = -1 ,

lim cth(x) = +1 .

9
5. Egy&eacute;b (nem elemi) f&uuml;ggv&eacute;nye
10
11
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