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Komposisi Fungsi

Kompoisi dari f dan g didefinisikan : (f o g)(x) = f [ g(x) ] dan (g o f)(x) = g [ f(x) ]
Jika digambarkan dalam diagram panah menjadi


Adapun penjelasan tentang tata cara menentukan hasil akhir dari komposisi fungsi akan diuraikan pada contoh soal berikut ini

01. Misalkan f = {(1, 4), (2, 3), (3, 1), (4, 2)} dan g = {(1, 2), (2, 4), (3, 1), (4, 3)}, maka tentukanlah :
(a) f o g                                   (b) g o f

Jawab
(a) f o g = f [ g ]
              = f [ (1, 2), (2, 4), (3, 1), (4, 3) ]
              = {(1, 2)?(2, 3), (2, 4)?(4, 2), (3, 1)?(1, 4), (4, 3)?(3, 1)}
              = {(1, 3), (2, 2), (3, 4), (4, 1)}
(b) g o f = g [ f ]
              = g [(1, 4), (2, 3), (3, 1), (4, 2) ]
              = {(1, 4)?(4, 3), (2, 3)?(3, 1), (3, 1)?(1, 2), (4, 2)?(2, 4)}
              = {(1, 3), (2, 1), (3, 2), (4, 4)}

02. Diketahui dua fungsi f(x) = 2x � 1 dan g(x) = x2 � 3x + 5. Tentukanlah hasil dari :
(a) (f o g)(x)                        (b) (g o f)(x)

Jawab
(a) (f o g)(x) = f [ g(x) ]
                     = f [x2 � 3x + 5]
                     = 2(x2 � 3x + 5) � 1
                     = 2x2 � 6x + 10 � 1
                     = 2x2 � 6x + 9
(b) (g o f)(x) = g [ f(x) ]
                     = g [2x � 1]
                     = (2x � 1)2 � 3(2x � 1) + 5
                     = 4x2 � 4x + 1 � 6x + 3 + 5
                     = 4x2 � 10x + 9

04. Diketahui fungsi f(x) = 2x + 3. Tentukanlah hasil dari (f o f o f)(x)
Jawab
(f o f o f)(x) = f { f [ f(x) ] }
                    = f { f [ 2x + 3 ] }
                    = f { 2(2x + 3) + 3] }
                    = f { 4x + 6 + 3 }
                    = f { 4x + 9 }
                    = 2(4x + 9) + 3
                    = 8x + 18 + 3
                    = 8x + 21

05. Diketahui dua fungsi f(x) = x2 � 5x + 4 dan g(x) = x2 + 3x �6. Tentukanlah nilai
      (a) (f o g)(2)                             (b) (g o f)(3)
Jawab
(a) (f o g)(2) = f [ g(2) ]
                     = f [(2)2 + 3(2) � 6]
                     = f [4 + 6 � 6]
                     = f [4]
                     = (4)2 � 5(4) + 4
                     = 16 � 20 + 4
                     = 0
(b) (g o f)(3) = g [ f(3) ]
                     = g [(3)2 � 5(3) + 4]
                     = g [9 � 15 + 4]
                     = g [�2]
                     = (�2)2 + 3(�2) � 6
                     = 4 � 6 � 6
                     = �8

Dari uraian di atas dapat ditentukan beberapa sifat komposisi fungsi, yakni
(1) Komposisi fungsi tidak komutatif, artinya : g o f ? f o g
(2) Komposisi fungsi bersifat asosiatif, artinya : f o [ g o h ] = [ f o g ] o h

Selanjutnya, kita dapat menentukan komponen fungsi komposisi jika hasil akhir komposisinya diketahui. Untuk penjelasan selengkapnya akan diuraikan pada contoh soal berikut ini :

06. Diketahui f(x) = 2x2 � 4x + 5 maka tentukanlah f(x + 3)
Jawab
f(x) = 2x2 � 4x + 5
maka f(x + 3) = 2(x + 3)2 � 4(x + 3) + 5
f(x + 3) = 2(x2 + 6x + 9) � 4x � 12 + 5
f(x + 3) = 2x2 + 12x + 18 � 4x � 12 + 5
f(x + 3) = 2x2 + 8x + 11

07. Diketahui f(x � 2) = x2 + 5x � 3 maka tentukanlah f(x)
Jawab
f(x � 2) = x2 + 5x � 3
Misalkan x � 2 = m maka x = m + 2
sehingga f(m) = (m + 2)2 + 5(m + 2) � 3
f(m) = m2 + 4m + 4 + 5m + 10 � 3
f(m) = m2 + 9m + 11
Jadi f(x) = x2 + 9x + 11

08. Diketahui f(2x + 3) = 4x2 � 8x + 5 maka tentukanlah f(x)
Jawab
f(2x + 3) = 4x2 � 8x + 5

09. Diketahui (f o h)(x) = 2x2 � 4x � 3 dan h(x) = x + 3 maka tentukanlah f(x)
Jawab
(f o h)(x) = 2x2 � 4x � 3
f [ h(x) ] = 2x2 � 4x � 3
f [x + 3] = 2x2 � 4x � 3
Karena x + 3 = h maka x = h � 3
sehingga f(h) = 2(h � 3)2 � 4(h � 3) � 3
f(h) = 2(h2 � 6h + 9) � 4(h � 3) � 3
f(h) = 2h2 � 12h + 18 � 4h + 12 � 3
f(h) = 2h2 � 16h + 27
Jadi f(x) = 2x2 � 16x + 27

10. Diketahui (f o h)(x) = 4x2 + 6x � 5 dan f(x) = 2x � 1 maka tentukanlah h(x)
Jawab
(f o h)(x) = 4x2 + 6x � 5
f [ h(x) ] = 4x2 + 6x � 5
2.h(x) � 1 = 2x2 � 4x � 3
2.h(x) = 2x2 � 4x � 3 + 1
2.h(x) = 2x2 � 4x � 2
h(x) = x2 � 2x � 1


12. Diketahui (f o g)(x) = 2x2 � 6x + 7 dan g(x) = x2 � 3x + 4 maka tentukanlah f(x)
Jawab
(f o g)(x) = 2x2 � 6x + 7
f [ g(x) ] = 2x2 � 6x + 7
f (x2 � 3x + 4) = 2x2 � 6x + 7

Misalkan m = x2 � 3x + 4 maka 2m = 2x2 � 6x + 8
2m � 1 = 2x2 � 6x + 8 � 1
2m � 1 = 2x2 � 6x + 7
sehingga f(m) = 2m � 1
Jadi f(x) = 2x � 1

13. Diketahui (f o g)(x) = 4x2 � 12x + 18 dan g(x) = x2 + 3x + 5, maka tentukanlah fungsi f(x)
Jawab
(f o g)(x) = 4x2 � 12x + 18
f [ g(x) ] = 4x2 � 12x + 18
f (x2 + 3x + 5) = 4x2 � 12x + 18

Misalkan m = x2 + 3x + 5 maka 4m = 4x2 + 12x + 20
4m � 2 = 2x2 � 6x + 20 � 2
4m � 2 = 2x2 � 6x + 18
sehingga f(m) = 4m � 2
Jadi f(x) = 4x � 2


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