Note;
@Jangan Asal Jawab
@Pake Cara
Fungsi Komposisi dan Fungsi Invers
Pendahuluan
A. Definisi Fungsi
Fungsi dari himpunan A ke Himpunan B => relasi yang memetakan setiap anggota A dengan tetap satu anggota B.
[tex] \: [/tex]
[tex] \boxed{\boxed{\mathbf{B.\ \ Operasi\ Aljabar}}}[/tex]
[tex] \scriptsize\boxed{\begin{array}{c}\mathbf{1.\ Penjumlahan\ dan\ Pengurangan\ Fungsi}\\\mathbf{\left(f\pm g\right)\left(x\right)=f\left(x\right)\pm g\left(x\right)}\\\\\mathbf{2.\ Perkalian\ Fungsi}\\\mathbf{\left(f\ .\ g\right)\left(x\right)=f\left(x\right)g\left(x\right)}\\\\\mathbf{3.\ Pembagian\ Fungsi}\\\mathbf{\left(\frac{f}{g}\right)\left(x\right)=\frac{f\left(x\right)}{g\left(x\right)}}\\\\\mathbf{4.\ Perpangkatan}\\\mathbf{\left(f\left(x\right)\right)^{n}=f^{n}\left(x\right)}\end{array}}[/tex]
[tex] \: [/tex]
[tex] \boxed{\boxed{\mathbf{C.\ \ Fungsi\ Komposisi}}}[/tex]
[tex] \scriptsize\mathbf{1.\ Fungsi\ komposisi\ dapat\ ditulis\ sebagai\ :}\\\\\mathbf{\left(f \circ g\right)\left(x\right)=f\left(g\left(x\right)\right)\to komposisi\ g}\\\mathbf{\left(g \circ f\right)\left(x\right)=g\left(f\left(x\right)\right)\to komposisi\ f}[/tex]
[tex] \boxed{\underbrace{\mathbf{x\to_{g}\ g\left(x\right)\to_{f}\ f\left(g\left(x\right)\right)}}_{\mathbf{\left(f\circ g\right)\left(x\right)=f\left(g\left(x\right)\right)}}} [/tex]
[tex] \: [/tex]
[tex]\boxed{\boxed{\mathbf{D. \ \ Fungsi \ Invers}}}[/tex]
[tex]\small\mathbf{1.) \ f^{-1} (x) \to invers\ dari\ fungsi\ f\left(x\right).} [/tex]
[tex]\boxed{\mathbf{\boxed{\mathbf{f^{-1}\left(y\right)=x}}\ _{f^{-1}} \rightleftharpoons ^{f} \ \boxed{\mathbf{y=f\left(x\right)}}}} [/tex]
[tex] \: [/tex]
[tex]\scriptsize\mathbf{2.) \ Invers\ dapat\ ditentukan\ dengan\ mengubah\ bentuk}[/tex]
[tex]\scriptsize\mathbf{f\left(x\right)=y=...} \ \scriptsize\mathbf{menjadi} \ \scriptsize\mathbf{f^{-1}\left(y\right)=x=...}[/tex]
[tex] \: [/tex]
[tex]\mathbf{3.)\ Sifat\ fungsi \ invers \ :} [/tex]
[tex]\mathbf{a.\ \left(f \circ f^{-1}\right)\left(x\right)=\left(f^{-1} \circ f\right)\left(x\right)=I\left(x\right)} [/tex]
[tex]\mathbf{b.\ \left(f \circ g\right)^{-1}\left(x\right)=\left(g^{-1} \circ f^{-1}\right)\left(x\right)} [/tex]
[tex]\mathbf{c.\ \left(f \circ g\right)\left(x\right)=h\left(x\right)\to f\left(x\right)=\left(h \circ g^{-1}\right)\left(x\right)} [/tex]
[tex] \: [/tex]
[tex]\mathbf{4.\ Rumus \ Cepat :}[/tex]
[tex]\small\boxed{\mathbf{f\left(x\right)=\frac{ax+b}{cx+d}\to f^{-1}\left(x\right)=\frac{-dx+b}{cx-a}}}[/tex]
[tex] \: [/tex]
[tex] \: [/tex]
Pembahasan
Nomor 9
Diketahui :
[tex]\bf{f\left(x\right)=\frac{2x}{3x-1}}[/tex]
[tex]\bf{g\left(x\right)=x-1}[/tex]
Ditanya :
[tex]\bf{\left(g \circ f\right)^{-1}\left(x\right)=...}[/tex]
Jawaban :
[tex]\bf{\left(g \circ f\right)\left(x\right)=g\left(f\left(x\right)\right)}[/tex]
[tex]\bf{\left(g \circ f\right)\left(x\right)=g\left(\frac{2x}{3x-1}\right)}[/tex]
[tex]\bf{\left(g \circ f\right)\left(x\right)=\left(\frac{2x}{3x-1}\right)-1}[/tex]
[tex]\bf{\left(g \circ f\right)\left(x\right)=\frac{2x-3x+1}{3x-1}}[/tex]
[tex]\bf{\left(g \circ f\right)\left(x\right)=\frac{-x+1}{3x-1}}[/tex]
maka (gunakan rumus cepat)
[tex]\bf{\left(g \circ f\right)^{-1}\left(x\right)=\frac{x+1}{3x+1}}[/tex]
Jadi jawabannya ialah
[tex]\boxed{\bf{a.\ \frac{x+1}{3x+1};x\ne-\frac{1}{3}}}[/tex]
[tex] \: [/tex]
Nomor 10
Diketahui :
[tex]\bf{f\left(x\right)=\frac{x-2}{x+2}}[/tex]
[tex]\bf{g\left(x\right)=x+2}[/tex]
Ditanya :
[tex]\bf{\left(f \circ g\right)^{-1}\left(x\right)=...}[/tex]
Jawaban :
[tex]\bf{\left(f \circ g\right)\left(x\right)=f\left(g\left(x\right)\right)}[/tex]
[tex]\bf{\left(f \circ g\right)\left(x\right)=f\left(x+2\right)}[/tex]
[tex]\bf{\left(f \circ g\right)\left(x\right)=\frac{\left(x+2\right)-2}{\left(x+2\right)+2}}[/tex]
[tex]\bf{\left(f \circ g\right)\left(x\right)=\frac{x-0}{x+4}}[/tex]
maka
[tex]\bf{\left(f \circ g\right)^{-1}\left(x\right)=\frac{-4x}{x-1}}[/tex]
Jadi jawabannya ialah
[tex]\boxed{\bf{a.\ \frac{-4x}{x-1}\ ;\ x\ne1}}[/tex]
[tex] \: [/tex]
[tex] \: [/tex]
Pelajari Lebih Lanjut :
- Contoh soal invers_Diketahui f(x) = x² dan g(x) = 4x -1. Jika h(x) = f(g(x) + 2) maka h^-1(x) adalah... : https://brainly.co.id/tugas/50517614
- Contoh soal Fungsi invers dari f(x) = 3x + 1: https://brainly.co.id/tugas/50517920
- Contoh soal Fungsi komposisi dan Fungsi Invers : https://brainly.co.id/tugas/50509104
- Contoh soal mencari fungsi komposisi -> (g o f) (x) : https://brainly.co.id/tugas/49941623
[tex] \: [/tex]
[tex] \: [/tex]
Detail Jawaban
Kelas : 11 SMA
Bab : 2
Sub Bab : Bab 6 - Fungsi
Kode Kategorisasi : 11.2.6
Kata Kunci : Fungsi Komposisi dan Fungsi invers.
[answer.2.content]
