Fungsi Komposisi : Pengertian, Rumus, Contoh Soal - Kalkulus
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Komposisi adalah salah satu metode untuk mengabungkan fungsi.
Definisi Jika f dan g adalah fungsi, fungsi komposisi f ◦ g (“f dikomposisikan dengan g”) didefinisikan sebagai
(f ◦ g)(x) = f(g(x))
Daerah asal f ◦ g terdiri dari bilangan x pada daerah asal g dengan g(x) berada di daerah asal f.
Jika ![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEitwtiHjBlC5_t8A2SwH_SiNKrlj1DGNB2oyFbBcV1WyOYrcj4Nm4zrufICNWioPJAlkqJiOZEGu7O3kdB-ZvYwrM3aOmz-VBzzfX84JWRuhrfFN35JcioFPSAAhD0Pk3DGczxpOW8FEIrf/s1600/CodeCogsEqn+-+2020-03-25T115615.664.gif)
Carilah!
a. f(g(x)) b. f(g(0))
c. g(f(x)) d. g(f(0))
e. f(f(x)) f. f(f(-5))
g. g(g(x)) h. g(g(2))
Jawab:
a. f(g(x)) = (x^2 - 3) + 5
= x^2 + 2
b. f(g(0)) = 0^2 + 2
= 2
c. g(f(x)) = (x + 5)^2 - 3
= (x^2 + 10x + 25) - 3
= x^2 + 10x + 22
d. g(f(0)) = 0^2 + 10.0 + 22
= 22
e. f(f(x)) = (x + 5) + 5
= x + 10
f. f(f(-5)) = -5 + 10
= 5
g. g(g(x)) = (x^2 - 3)^2 - 3
= (x^4 - 6x^2 + 9) - 3
= x^4 - 6x^2 + 6
h. g(g(2)) = 2^4 - 6.2^2 + 6
= 16 - 24 + 6
= -2
Contoh 1.2
Tulislah rumus untuk f ◦ g ◦ h
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgr2Mv7c3UE9yIf3HFtlZ4_zyAgqu4UGakL5a7BmG3o6OiyrlzUpLGzf-Km1QtS5dF2HfwrTkvsKsYdrAcDNQ076n_uFU1Pm1XX388cCmEc00FxF7d4vavBjEkmRngMZi1rP_Qjgp-TQZuG/s640/CodeCogsEqn+-+2020-03-25T115922.902.gif)
Jawab:
(f ◦ g)(x) = f(g(x))
= 3x + 1
((f ◦ g) ◦ h)(x) = (f ◦ g)(h(x))
= 3.(4 - x) + 1
= 12 - 3x + 1
= 13 - 3x
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjOGBEj7BlpgVzCmsj5-Luw3akGqTh0OvD888ewGLfHLlM_y0WgDSjUwOSEwowSjE8AJbNNvOfv9itTgiJHMdSXDWw58yJ9oQJZ0Sm2-QzT0dC5U9AXEokaM5Te2H-LoBmaSwiFvWbShPce/s1600/CodeCogsEqn+-+2020-03-25T120324.137.gif)
(f ◦ g)(x) = f(g(x))
= 3.(2x - 1) + 4
= 6x - 3 + 4
= 6x + 1
((f ◦ g) ◦ h)(x) = (f ◦ g)(h(x))
= 6.(x^2) + 1
= 6x^2 + 1
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhgH9EyLLabIb_CrcGC-9yeer6f1pmNTI48y4sVOBjJufZJyphb2Yv9uuZDz8uDw5uDRx9BCA6wns1CCAr_BU7-NGY_Jc_ikvDSmbB16lPprpGG1_oZE68UHOccxTsMYgLCxE6be6lD6Ni1/s1600/CodeCogsEqn+-+2020-03-25T120401.058.gif)
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjNmoEOlJcfhdV6R4ZMaECbH_XaWxGSR-Q_7__M1oEKllPBXSs5OtQZB4zUbCk_T0wQ5jT8l8cAIRo7QG29XJYxUsyap6ifHyTtGHo853Crmb5L-rq9h5LXIn5ejK20ZLy2v3SSn1rKG8y4/s640/CodeCogsEqn+%25283%2529.gif)
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh7LvsbbABrF_-RgEW52dLdJci5Z7lrANSp9o34_xoxU26h_zFJIoFUgLdHwDnfuDMdUAvoWkSB2mwfKDdzAXsjBhnvcb82IDo7TGyoYldlHAbYLpN-Nhyphenhyphene3xsMopLtbUobtOnGu1b7sQFz/s1600/CodeCogsEqn+-+2020-03-25T121110.216.gif)
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEitwtiHjBlC5_t8A2SwH_SiNKrlj1DGNB2oyFbBcV1WyOYrcj4Nm4zrufICNWioPJAlkqJiOZEGu7O3kdB-ZvYwrM3aOmz-VBzzfX84JWRuhrfFN35JcioFPSAAhD0Pk3DGczxpOW8FEIrf/s1600/CodeCogsEqn+-+2020-03-25T115615.664.gif)
Carilah!
a. f(g(x)) b. f(g(0))
c. g(f(x)) d. g(f(0))
e. f(f(x)) f. f(f(-5))
g. g(g(x)) h. g(g(2))
a. f(g(x)) = (x^2 - 3) + 5
= x^2 + 2
b. f(g(0)) = 0^2 + 2
= 2
c. g(f(x)) = (x + 5)^2 - 3
= (x^2 + 10x + 25) - 3
= x^2 + 10x + 22
d. g(f(0)) = 0^2 + 10.0 + 22
= 22
e. f(f(x)) = (x + 5) + 5
= x + 10
f. f(f(-5)) = -5 + 10
= 5
g. g(g(x)) = (x^2 - 3)^2 - 3
= (x^4 - 6x^2 + 9) - 3
= x^4 - 6x^2 + 6
h. g(g(2)) = 2^4 - 6.2^2 + 6
= 16 - 24 + 6
= -2
Contoh 1.2
Tulislah rumus untuk f ◦ g ◦ h
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgr2Mv7c3UE9yIf3HFtlZ4_zyAgqu4UGakL5a7BmG3o6OiyrlzUpLGzf-Km1QtS5dF2HfwrTkvsKsYdrAcDNQ076n_uFU1Pm1XX388cCmEc00FxF7d4vavBjEkmRngMZi1rP_Qjgp-TQZuG/s640/CodeCogsEqn+-+2020-03-25T115922.902.gif)
Jawab:
(f ◦ g)(x) = f(g(x))
= 3x + 1
((f ◦ g) ◦ h)(x) = (f ◦ g)(h(x))
= 3.(4 - x) + 1
= 12 - 3x + 1
= 13 - 3x
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjOGBEj7BlpgVzCmsj5-Luw3akGqTh0OvD888ewGLfHLlM_y0WgDSjUwOSEwowSjE8AJbNNvOfv9itTgiJHMdSXDWw58yJ9oQJZ0Sm2-QzT0dC5U9AXEokaM5Te2H-LoBmaSwiFvWbShPce/s1600/CodeCogsEqn+-+2020-03-25T120324.137.gif)
(f ◦ g)(x) = f(g(x))
= 3.(2x - 1) + 4
= 6x - 3 + 4
= 6x + 1
((f ◦ g) ◦ h)(x) = (f ◦ g)(h(x))
= 6.(x^2) + 1
= 6x^2 + 1
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhgH9EyLLabIb_CrcGC-9yeer6f1pmNTI48y4sVOBjJufZJyphb2Yv9uuZDz8uDw5uDRx9BCA6wns1CCAr_BU7-NGY_Jc_ikvDSmbB16lPprpGG1_oZE68UHOccxTsMYgLCxE6be6lD6Ni1/s1600/CodeCogsEqn+-+2020-03-25T120401.058.gif)
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjNmoEOlJcfhdV6R4ZMaECbH_XaWxGSR-Q_7__M1oEKllPBXSs5OtQZB4zUbCk_T0wQ5jT8l8cAIRo7QG29XJYxUsyap6ifHyTtGHo853Crmb5L-rq9h5LXIn5ejK20ZLy2v3SSn1rKG8y4/s640/CodeCogsEqn+%25283%2529.gif)
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiTGZexIpJ0wwOnKRdg6673xIj_wAI7QhvK4pz5BcDpRqy90y_Ms312zI0EFqSlvk1W3o5kToXFclffScDu55DCciG31E4iATuPqadsNMfak6vY7FeWqgd_JENKzaJ9-m_1_6j470uTkf04/s640/CodeCogsEqn+-+2020-03-25T120453.863.gif)
Buat Latihan. Saya kasih jawaban yang benar:
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh7LvsbbABrF_-RgEW52dLdJci5Z7lrANSp9o34_xoxU26h_zFJIoFUgLdHwDnfuDMdUAvoWkSB2mwfKDdzAXsjBhnvcb82IDo7TGyoYldlHAbYLpN-Nhyphenhyphene3xsMopLtbUobtOnGu1b7sQFz/s1600/CodeCogsEqn+-+2020-03-25T121110.216.gif)