Fungsi Onto
• Fungsi f dikatakan onto atau surjektif jika setiap
elemen himpunan C merupakan bayangan dari satu
atau lebih elemen himpunan A.
• Dengan kata lain seluruh elemen C merupakan
range dari f
• Fungsi f disebut fungsi pada himpunan COnto ? One to One ?
a
b
c
1
2
3
4
a
b
c
d
1
2
3Onto ? One to One ?
a
b
c
d
1
2
3
4
a
b
c
d
1
2
3
4Jawaban
• One to one
• Onto
• One to one dan Onto
• Bukan One to one dan OntoFungsi Bijektif (Bijective)
Fungsi f : X→ Y bijektif ⇔
f adalah one-to-one
dan onto
–Contoh
• Fungsi linear f(x) = ax + b
• Fungsi f(x) = x
Fungsi Invers
• Terdapat sebuah fungsi y =f(x), fungsi
inverse f –1
adalah himpunan {(y,x) | y
=f(x)}.
• Fungsi one to one sering dinamakan
juga fungsi invertible (dapat dibalikkan)
• Fungsi bukan one to one sering
dinamakan juga fungsi not invertible
(tidak dapat dibalikkan)Inversion
(= invertible functions)Inversion
•f
-1
:C→P is no
function, because it is
not defined for all
elements of C and
assigns two images to
the pre-image New
York.Contoh :
• Daerah asal A = {1,2,3}
• Daerah asal B = {u,v,w}
• F = {(1,u),(2,w),(3,v)}
– Adalah fungsi one-to-one
– Invers f
-1
= {(u,1),(w,2),(v,3)}Contoh :
• Tentukan invers dari f(x) = x – 1
• Fungsi f(x) = x – 1 adalah fungsi one to
one, jadi mempunyai invers
• f(x) = x – 1
• y = x – 1
• x = y + 1
• Maka fungsi invers f
-1
(x) = x + 1Contoh
• Tentukan fungsi invers f(x) = x
2
+ 1
• Fungsi f(x) = x
2
+ 1 bukan fungsi one to
one sehingga tidak memiliki fungsi inversKomposisi fungsi
• Terdapat dua fungsi g : X → Y dan f : Y → Z,
komposisi f ◦ g didefinisikan sebagai berikut :
• f ◦ g (x) = f(g(x)) untuk setiap x ∈ X
• Contoh :
– g(x) = x
2
-1,
– f(x) = 3x + 5.
– g ◦ f(x) = g(f(x)) = g(3x + 5) = (3x + 5)
2
- 1
• Komposisi fungsi memenuhi hokum
assosiatif : f ◦ (g ◦h) = (f ◦ g) ◦ h tetapi tidak
memenuhi hukum komutatif : f ◦ g ≠ g ◦ f.Composition
•The composition of two functions g:B→C and f:C→D,
denoted by f°g, is defined by
•(f°g)(a) = f(g(a))
•This means that
• first, function g is applied to element a∈B,
mapping it onto an element of C,
• then, function f is applied to this element of
C, mapping it onto an element of D.
• Therefore, the composite function maps
from B to D.Composition
•Example:
•f(x) = 7x – 4, g(x) = 3x,
•f:R→R, g:R→R
•(f°g)(5) = f(g(5)) = f(15) = 105 – 4 = 101
•(f°g)(x) = f(g(x)) = f(3x) = 21x - 4Composition
•Composition of a function and its inverse:
•(f
-1
°f)(x) = f
-1
(f(x)) = x
•The composition of a function and its inverse is the
identity function i(x) = x.Graphs
•The graph of a function f:A→B is the set of ordered
pairs {(a, b) | a∈A and f(a) = b}.
•The graph is a subset of A×B that can be used to
visualize f in a two-dimensional coordinate system.Floor and Ceiling Functions
•The floor and ceiling functions map the real numbers onto
the integers (R→Z).
•The floor function assigns to r∈R the largest z∈Z with z≤r,
denoted by ⎣r⎦.
•Examples: ⎣2.3⎦ = 2, ⎣2⎦ = 2, ⎣0.5⎦ = 0, ⎣-3.5⎦ = -4
•The ceiling function assigns to r∈R the smallest z∈Z with
z≥r, denoted by ⎡r⎤.
•Examples: ⎡2.3⎤ = 3, ⎡2⎤ = 2, ⎡0.5⎤ = 1, ⎡-3.5⎤ = -3
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