Elementar funksiyalar — koʻphadlar, ratsional , koʻrsatkichli, darajali, logarifmik, trigonometrik, teskari trigonometrik funksiyalar, shuningdek, bu funksiyalardan toʻrt arifmetik amal va chekli marta qoʻllangan superpozitsiyalar yordamida hosil qilinadigan funksiyalarni oʻz ichiga olgan funksiyalar sinfi.
Elementar funksiyalar sinfi yaxshi oʻrganilgan va u amaliy matematikada koʻp qoʻllanadi. Elementar funksiyalar ning hosilasi hamisha Elementar funksiyalar boʻladi, lekin Elementar funksiyalardan olingan integral Elementar funksiyalar boʻlmasligi ham mumkin.
Ushbu
y
=
a
0
+
a
1
x
+
a
2
x
2
+
.
.
.
a
n
−
1
x
n
−
1
+
a
n
x
n
{\displaystyle y=a_{0}+a_{1}x+a_{2}{x}^{2}+...a_{n-1}x^{n-1}+a_{n}x^{n}}
a
0
,
a
1
,
.
.
.
,
a
n
{\displaystyle a_{0},a_{1},...,a_{n}}
n
∈
N
{\displaystyle n\in N}
R
=
(
−
∞
,
+
∞
)
{\displaystyle R=(-\infty ,+\infty )}
Bu funksiya
y
=
a
x
+
b
{\displaystyle y=ax+b}
(
a
≠
0
)
{\displaystyle (a\neq 0)}
Chiziqli funksiya
(
−
∞
,
+
∞
)
{\displaystyle (-\infty ,+\infty )}
a
>
0
{\displaystyle a>0}
a
<
0
{\displaystyle a<0}
Parabolaning tekislikda joylashishi
a
{\displaystyle a}
D
=
b
2
−
4
a
c
{\displaystyle D=b^{2}-4ac}
a
>
0
{\displaystyle a>0}
D
>
0
{\displaystyle D>0}
a
<
0
{\displaystyle a<0}
D
<
0
{\displaystyle D<0}
Bu funksiya
y
=
a
x
2
+
b
x
+
c
{\displaystyle y=ax^{2}+bx+c}
(
a
≠
0
)
{\displaystyle (a\neq 0)}
Kvadrat funktsiya R da aniqlangan boʼlib, uning grafigi parabolani ifodalaydi.
Ravshanki
y
=
a
x
2
+
b
x
+
c
=
a
(
x
+
b
2
a
)
−
b
2
−
4
a
c
4
a
{\displaystyle y=ax^{2}+bx+c=a{\Bigl (}x+{\frac {b}{2a}}{\Bigr )}-{\frac {b^{2}-4ac}{4a}}}
Ushbu
y
=
a
0
+
a
1
x
+
a
2
x
2
+
.
.
.
+
a
n
x
n
b
0
+
b
1
x
+
b
2
x
2
+
.
.
.
+
b
m
x
m
{\displaystyle y={\frac {a_{0}+a_{1}x+a_{2}x^{2}+...+a_{n}x^{n}}{b_{0}+b_{1}x+b_{2}x^{2}+...+b_{m}x^{m}}}}
a
0
,
a
1
,
.
.
.
,
a
n
{\displaystyle a_{0},a_{1},...,a_{n}}
b
0
,
b
1
,
.
.
.
,
b
m
{\displaystyle b_{0},b_{1},...,b_{m}}
n
∈
N
{\displaystyle n\in N}
m
∈
N
{\displaystyle m\in N}
X
=
(
−
∞
,
+
∞
)
{
x
∣
b
0
+
b
1
x
+
.
.
.
+
b
m
x
m
=
0
{\displaystyle X=(-\infty ,+\infty )\ \{x\mid b_{0}+b_{1}x+...+b_{m}x^{m}=0}
Kasr ratsional funksiyaning baʻzi xususiy hollari:
Bu funksiya
X
=
(
−
∞
,
0
)
⋃
(
0
,
+
∞
)
=
R
∖
{
0
}
{\displaystyle X=(-\infty ,0)\bigcup (0,+\infty )=R\backslash \{0\}}
a
{\displaystyle a}
(
−
∞
,
0
)
{\displaystyle (-\infty ,0)}
(
0
,
+
∞
)
{\displaystyle (0,+\infty )}
U
y
=
a
x
{\displaystyle y={\frac {a}{x}}}
x
≠
0
{\displaystyle x\neq 0}
a
=
c
o
n
s
t
{\displaystyle a=const}
U ushbu
y
=
a
x
+
b
c
x
+
d
{\displaystyle y={\frac {ax+b}{cx+d}}}
X
=
R
∖
{
−
d
c
}
{\displaystyle X=R\backslash \{-{\frac {d}{c}}\}}
(
c
≠
0
)
{\displaystyle (c\neq 0)}
Ravshanki,
y
=
a
x
+
b
c
x
+
d
=
b
c
−
a
d
c
2
∗
1
x
+
d
c
+
a
c
{\displaystyle y={\frac {ax+b}{cx+d}}={\frac {bc-ad}{c^{2}}}*{\frac {1}{x+{\frac {d}{c}}}}+{\frac {a}{c}}}
y
=
a
x
+
β
+
γ
{\displaystyle y={\frac {a}{x+\beta }}+\gamma }
(
a
=
b
c
−
a
d
c
2
,
β
=
d
c
,
γ
=
a
c
)
{\displaystyle {\Bigr (}a={\frac {bc-ad}{c^{2}}},\beta ={\frac {d}{c}},\gamma ={\frac {a}{c}}{\Bigr )}}
y
=
a
x
{\displaystyle y={\frac {a}{x}}}
Ushbu
y
=
x
a
{\displaystyle y=x^{a}}
(
x
≥
0
)
{\displaystyle (x\geq 0)}
Bu funksiyaning aniqlanish toʻplami
a
{\displaystyle a}
a
>
0
{\displaystyle a>0}
(
0
,
+
∞
)
{\displaystyle (0,+\infty )}
a
<
0
{\displaystyle a<0}
y
=
x
a
{\displaystyle y=x^{a}}
Ushbu
y
=
a
x
{\displaystyle y=a^{x}}
a
∈
R
{\displaystyle a\in R}
a
>
0
{\displaystyle a>0}
a
≠
1
{\displaystyle a\neq 1}
(
−
∞
,
+
∞
)
{\displaystyle (-\infty ,+\infty )}
∀
x
∈
R
{\displaystyle \forall x\in R}
a
x
>
0
{\displaystyle a^{x}>0}
a
>
0
{\displaystyle a>0}
0
<
a
<
1
{\displaystyle 0<a<1}
Xususan,
a
=
e
{\displaystyle a=e}
y
=
e
x
{\displaystyle y=e^{x}}
Koʻrsatkichli funksiyaning grafigi
O
x
{\displaystyle Ox}
Ushbu
y
=
log
y
x
{\displaystyle y=\log _{y}x}
a
>
0
{\displaystyle a>0}
a
≠
1
{\displaystyle a\neq 1}
Logarifimlik funksiya
(
0
,
+
∞
)
{\displaystyle (0,+\infty )}
y
=
a
x
{\displaystyle y=a^{x}}
a
>
1
{\displaystyle a>1}
0
<
a
<
1
{\displaystyle 0<a<1}
Logarifmik funksiyaning grafigi
O
y
{\displaystyle Oy}
Ushbu
y
=
sin
x
{\displaystyle y=\sin x}
y
=
cos
x
{\displaystyle y=\cos x}
y
=
t
g
x
{\displaystyle y=tgx}
y
=
c
t
g
x
{\displaystyle y=ctgx}
y
=
s
e
c
x
{\displaystyle y=secx}
y
=
c
o
s
e
c
x
{\displaystyle y=cosecx}
y
=
sin
x
{\displaystyle y=\sin x}
y
=
cos
x
{\displaystyle y=\cos x}
R
=
(
−
∞
,
+
∞
)
{\displaystyle R=(-\infty ,+\infty )}
2
π
{\displaystyle 2\pi }
∀
x
∈
R
{\displaystyle \forall x\in R}
−
1
≤
s
i
n
x
≤
1
{\displaystyle -1\leq sinx\leq 1}
−
1
≤
c
o
s
x
≤
1
{\displaystyle -1\leq cosx\leq 1}
y
=
t
g
x
{\displaystyle y=tgx}
X
=
R
∖
{
x
∈
R
|
x
=
(
2
k
+
1
)
π
2
;
k
=
0
,
±
1
,
±
2
,
.
.
.
}
{\displaystyle X=R\backslash \{x\in R|x=(2k+1){\frac {\pi }{2}};k=0,\pm 1,\pm 2,...\}}
π
{\displaystyle \pi }
c
t
g
x
{\displaystyle ctgx}
s
e
c
x
{\displaystyle secx}
c
o
s
e
c
x
{\displaystyle cosecx}
s
i
n
x
{\displaystyle sinx}
c
o
s
x
{\displaystyle cosx}
t
g
x
{\displaystyle tgx}
c
t
g
x
=
1
t
g
x
,
s
e
c
x
=
1
c
o
s
x
,
c
o
s
e
c
x
=
1
s
i
n
x
{\displaystyle ctgx={\frac {1}{tgx}},secx={\frac {1}{cosx}},cosecx={\frac {1}{sinx}}}
Koʻrsatkichli
y
=
e
x
{\displaystyle y=e^{x}}
e
x
−
e
−
x
2
,
e
x
+
e
−
x
2
,
e
x
−
e
−
x
e
x
+
e
−
x
,
e
x
+
e
−
x
e
x
−
e
−
x
{\displaystyle {\frac {e^{x}-e^{-x}}{2}},{\frac {e^{x}+e^{-x}}{2}},{\frac {e^{x}-e^{-x}}{e^{x}+e^{-x}}},{\frac {e^{x}+e^{-x}}{e^{x}-e^{-x}}}}
s
h
x
=
e
x
−
e
−
x
2
,
c
h
x
=
e
x
+
e
−
x
2
,
t
h
x
=
e
x
−
e
−
x
e
x
+
e
−
x
,
c
t
h
x
=
e
x
+
e
−
x
e
x
−
e
−
x
{\displaystyle shx={\frac {e^{x}-e^{-x}}{2}},chx={\frac {e^{x}+e^{-x}}{2}},thx={\frac {e^{x}-e^{-x}}{e^{x}+e^{-x}}},cthx={\frac {e^{x}+e^{-x}}{e^{x}-e^{-x}}}}
Maʻlumki,
y
=
s
i
n
x
{\displaystyle y=sinx}
Y
f
=
[
−
1
,
1
]
{\displaystyle Y_{f}=[-1,1]}
Agar
x
∈
(
−
π
2
)
(
π
2
)
{\displaystyle x\in \left(-{\frac {\pi }{2}}\right)\left({\frac {\pi }{2}}\right)}
X
=
(
−
π
2
)
(
π
2
)
{\displaystyle X=\left(-{\frac {\pi }{2}}\right)\left({\frac {\pi }{2}}\right)}
Y
f
=
[
−
1
,
1
]
{\displaystyle Y_{f}=[-1,1]}
y
=
s
i
n
x
{\displaystyle y=sinx}
y
=
a
r
c
s
i
n
x
{\displaystyle y=arcsinx}
Shunga oʻxshash
y
=
c
o
s
x
,
y
=
t
g
x
,
y
=
c
t
g
x
{\displaystyle y=cosx,y=tgx,y=ctgx}
y
=
a
r
c
c
o
s
x
,
y
=
a
r
c
t
g
x
,
y
=
a
r
c
c
t
g
x
{\displaystyle y=arccosx,y=arctgx,y=arcctgx}
Ushbu
y
=
a
r
c
s
i
n
x
,
y
=
a
r
c
c
o
s
x
,
y
=
a
r
c
t
g
x
,
y
=
a
r
c
c
t
g
x
{\displaystyle y=arcsinx,y=arccosx,y=arctgx,y=arcctgx}
[ 1]
↑ Гулмирза Худойберганов, Азизжон Ворисов. Элементар функциялар , МАТЕМАТИК АНАЛИЗДАН МАЪРУЗАЛАР, 2010 — 67-70-bet.
![{\displaystyle Y_{f}=[-1,1]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/63cdaa4e874fa77680d9c96244ce889c4edc0c8c)
