RelasiRelasi dari A ke B disebut fungsi apabila setiapelemen himpunan A dipasangkan hanya satu kali padaelemen himpunan By= f(x) ; artinya y merupakan fungsi xA = daerah asal (Domain)B = daerah jelajah (Kodomain)Relation from A to B is called a function if every elements of set A is paired only once in the elements of set B y = f (x), meaning that y is a function of x A = area of origin (Domain) B = the home range (Kodomain)

ABABAxAxByByCzCz

FUNGSI Bukan Fungsi

Komposisi Fungsi Jika fungsi f: A B dilanjutkan fungsi g: B C maka dapat dinyatakan dengan (g o f) : A CRumus : fg(i) (fog)(x) = f(g(x))AB C(ii) (gof)(x) = g(f(x))Xg(x)g(f(x))

gof

Contoh:f(x 1) = x2 + 5x Tentukan : a. f(g(x)) b. f(-3)Misal f(x) = x 1 maka g(x) = y + 1 karena f(x 1) = x2 + 5x maka f(g(x)) = (y + 1)2 + 5(y + 1) f(g(x)) = y2 + 2y + 1 + 5y + 5 f(g(x)) = y2 + 7y + 6

Example: f (x - 1) = x2 + 5x Define: a. f (g (x)) b. f (-3) Suppose f (x) = x - 1 then g (x) = y + 1 because f (x - 1) = x2 + 5x then f (g (x)) = (y + 1) 2 + 5 (y + 1) f (g (x)) = y2 + 2y + 1 + 5Y + 5 f (g (x)) = y2 + 7y + 6

a. fog = x2 + 7x + 6 b. fog(-1) = (-3)2 + 7(-3) + 6 = 9 21 + 6 = -6

Komposisi FungsiPenggabungan operasi dua fungsisecara berurutan akanmenghasilkan sebuah fungsi baru.Penggabungan tersebut disebutkomposisi fungsi dan hasilnyadisebut fungsi komposisi.

Merging the two functions operating will sequentially produce a new function. The merger has been called composition and function results called the composition function.

x A dipetakan oleh f ke y Bditulis f : x y atau y = f(x)y B dipetakan oleh g ke z Cditulis g : y z atau z = g(y)atau z = g(f(x))

fg

x A mapped byf to y Bwritten f : x y or y = f(x)y B mapped by g to z Cwritten g : y z or z = g(y)or z = g(f(x))

fg

maka fungsi yang memetakanx A ke z Cadalah komposisi fungsi f dan gditulis (g o f)(x) = g(f(x))

maka fungsi yang memetakanx A ke z Cadalah komposisi fungsi f dan gditulis (g o f)(x) = g(f(x)) then the function which mapsx A to z Cis the composition of functions f and g written g o f)(x) = g(f(x))

* Jawab:f(a) = 1 dan g(1) = qJadi (g o f)(a) = g(f(a)) = g(1) q(g o f)(a) = ?

f(b) = 3 dan g(3) = pJadi (g o f) = g(f(b)) = g(3) = p(g o f)(b) = ?

contoh 2

Ditentukan g(f(x)) = f(g(x)).Jika f(x) = 2x + p dang(x) = 3x + 120maka nilai p = .

Jawab:f(x) = 2x + p dan g(x) = 3x + 120g(f(x)) = f(g(x))g(2x+ p) = f(3x + 120)3(2x + p) + 120 = 2(3x + 120) + p6x + 3p + 120 = 6x + 360 + p3p p = 360 1202p = 240 p = 120

Sifat Komposisi FungsiTidak komutatif: f o g g o f2. Bersifat assosiatif:f o (g o h) = (f o g) o h = f o g o h3. Memiliki fungsi identitas: I(x) = xf o I = I o f = f

contoh 1f : R R dan g : R Rf(x) = 3x 1 dan g(x) = 2x2 + 5Tentukan: a. (g o f)(x) b. (f o g)(x)

Jawab:f(x) = 3x 1 dan g(x) = 2x2 + 5 (g o f)(x) = g[f(x)] = g(3x 1) = 2(3x 1)2 + 5 = 2(9x2 6x + 1) + 5 = 18x2 12x + 2 + 5 = 18x2 12x + 7

f(x) = 3x 1 dan g(x) = 2x2 + 5b. (f o g)(x) = f[g(x)] = f(2x2 + 5) = 3(2x2 + 5) 1 = 6x2 + 15 1 (f o g)(x) = 6x2 + 14 (g o f)(x) = 18x2 12x + 7 (g o f)(x) (f o g )(x) tidak bersifat komutatif

contoh 2f(x) = x 1, g(x) = x2 1 dan h(x) = 1/xTentukan: a. (f o g) o h b. f o (g o h)

Jawab:f(x) = x 1, g(x) = x2 1dan h(x) = 1/x((f o g) o h)(x) = (f o g)(h(x))(f o g)(x) = (x2 1) 1 = x2 2(f o g(h(x))) = (f o g)(1/x) = (1/x)2 2

f(x) = x 1, g(x) = x2 1dan h(x) = 1/x(f o (g o h))(x) = (f(g oh)(x))(g o h)(x)= g(1/x) = (1/x)2 1 = 1/x2 - 1f(g o h)(x)= f(1/x2 1) = (1/x2 1) 1 =(1/x)2 2

Contoh 4

Diketahui f(x) = 2x + 1dan (f o g)(x + 1)= -2x2 4x + 1Nilai g(-2) =.

Jawaban:f(g(x + 1))= -2x2 4x + 1f(x) = 2x + 1 f(g(x))= 2g(x) + 1f(g(x + 1)) = 2g (x + 1) + 12g(x + 1) + 1 = -2x2 4x 12g(x + 1) = -2x2 4x 2g(x + 1) = -x2 2x 1

g(x + 1) = -x2 2x 1 g(x) = -(x 1)2 2(x 1) 1 g(2) = -(2 1)2 2(2 1) 1 = -1 2 1 = -4Jadi g(2) = - 4

f(x) = x 1, g(x) = x2 1dan h(x) = 1/x(f o (g o h))(x) = (f(g oh)(x))(g o h)(x)= g(1/x) = (1/x)2 1 = 1/x2 - 1f(g o h)(x)= f(1/x2 1) = (1/x2 1) 1 =(1/x)2 2

Fungsi inversUntuk menentukan fungsi invers dari suatu fungsi dapat dilakukan dengan caraberikut ini.a. Buatlah permisalan f(x) = y pada persamaan.b. Persamaan tersebut disesuaikan dengan f(x) = y, sehingga ditemukan fungsi dalamy dan nyatakanlah x = f(y).c. Gantilah y dengan x, sehingga f(y) = f 1(x).Untuk lebih memahami tentang fungsi invers, pelajarilah contoh soal berikut ini.

To determine the function of the inverse of a function can be done by following. a. Make permisalan f (x) = y in the equation. b. This equation is adjusted to f (x) = y, so that is found in functions y and state x = f (y). c. Replace y with x, so f (y) = f -1 (x). To better understand the inverse function, learn about the following examples.

Jika diketahui f(x) = x x 2, tentukan inversnya X+2Misal f(x) = y, maka soalnya menjadi:F(x)= xX+2 y= xX+2X= -2yy(x + 2) = x y-1yx + 2y = x f1(x)= -2xyx x = 2yx-1(y 1)x = 2y

If known f(x) = x x 2, determine the inverse X+2Example f(x) = y, then the question becomes :F(x)= xX+2 y= xX+2X= -2yy(x + 2) = x y-1yx + 2y = x f1(x)= -2xyx x = 2yx-1(y 1)x = 2y

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