tolong kak jan ngasal.... Question from @wildaaaaa14

wildaaaaa14 @wildaaaaa14

March 2022 1 8 Report

tolong kak jan ngasal

Sinogen

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.

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$\boxed{\boxed{\mathbf{B.\ \ Fungsi\ Komposisi}}}$

$\scriptsize\mathbf{1.\ Fungsi\ komposisi\ dapat\ ditulis\ sebagai\ :}\\\\\mathbf{\left(f \circ g\right)\left(x\right)=f\left(g\left(x\right)\right)\to komposisi\ g}\\\mathbf{\left(g \circ f\right)\left(x\right)=g\left(f\left(x\right)\right)\to komposisi\ f}$

$\boxed{\underbrace{\mathbf{x\to_{g}\ g\left(x\right)\to_{f}\ f\left(g\left(x\right)\right)}}_{\mathbf{\left(f\circ g\right)\left(x\right)=f\left(g\left(x\right)\right)}}}$

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$\scriptsize\mathbf{2.\ Sifat\ fungsi\ komposisi,\ antara\ lain\ :}\\\\\mathbf{a.\ Tidak\ komutatif,\ \left(f \circ g\right)\left(x\right)\ne\left(g \circ f\right)\left(x\right).}\\\mathbf{b.\ Asosiatif,\ \left(f \circ \left(g \circ h\right)\right)\left(x\right)=\left(\left(f \circ g\right) \circ h\right)\left(x\right).}\\\mathbf{c.\ Terdapat\ unsur\ identitas\ \left(I\right)\ \left(x\right),\ }\\\mathbf{\left(f \circ I\right)\left(x\right)=\left(I \circ f\right)\left(x\right)=f\left(x\right).}$

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$\boxed{\boxed{\mathbf{C. \ \ Fungsi \ Invers}}}$

$\small\mathbf{1.) \ f^{-1} (x) \to invers\ dari\ fungsi\ f\left(x\right).}$

$\boxed{\mathbf{\boxed{\mathbf{f^{-1}\left(y\right)=x}}\ _{f^{-1}} \rightleftharpoons ^{f} \ \boxed{\mathbf{y=f\left(x\right)}}}}$

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$\scriptsize\mathbf{2.) \ Invers\ dapat\ ditentukan\ dengan\ mengubah\ bentuk}$

$\scriptsize\mathbf{f\left(x\right)=y=...} \ \scriptsize\mathbf{menjadi} \ \scriptsize\mathbf{f^{-1}\left(y\right)=x=...}$

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$\mathbf{3.)\ Sifat\ fungsi \ invers \ :}$

$\mathbf{a.\ \left(f \circ f^{-1}\right)\left(x\right)=\left(f^{-1} \circ f\right)\left(x\right)=I\left(x\right)}$

$\mathbf{b.\ \left(f \circ g\right)^{-1}\left(x\right)=\left(g^{-1} \circ f^{-1}\right)\left(x\right)}$

$\mathbf{c.\ \left(f \circ g\right)\left(x\right)=h\left(x\right)\to f\left(x\right)=\left(h \circ g^{-1}\right)\left(x\right)}$

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$\mathbf{4.\ Rumus \ Cepat :}$

$\small\boxed{\mathbf{f\left(x\right)=\frac{ax+b}{cx+d}\to f^{-1}\left(x\right)=\frac{-dx+b}{cx-a}}}$

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Pembahasan

Nomor 1

Jika f(x) = 5 - x dan g(x) = 2x + 3, maka nilai (g o f) invers adalah...

Jawaban :

$\mathbf{\left(g \circ f\right)\left(x\right)}$

$\mathbf{\left(g \circ f\right)\left(x\right)=g\left(f\left(x\right)\right)}$
$\mathbf{\left(g \circ f\right)\left(x\right)=g\left(5-x\right)}$
$\mathbf{\left(g \circ f\right)\left(x\right)=2\left(5-x\right)+3}$
$\mathbf{\left(g \circ f\right)\left(x\right)=13-2x}$

$\mathbf{\left(g \circ f\right)^{-1}\left(x\right)=?}$
_____________
$\mathbf{y=13-2x}$
$\mathbf{2x=13-y}$
$\mathbf{x=\frac{13-y}{2}}$

$\boxed{\mathbf{\left(g \circ f\right)^{-1}\left(x\right)=\frac{13-x}{2}}}$

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Nomor 2

Diketahui f(x) = ax + b, jika f(-3) = -11 dan f(7) = 9. Tentukan nilai a dan b...

Jawaban :

$\mathbf{f\left(x\right)=ax+b}$
___________

$\mathbf{f\left(-3\right)=-11}$
$\mathbf{-3a+b=-11...\left(1\right)}$

$\mathbf{f\left(7\right)=9}$
$\mathbf{7a+b=9...\left(2\right)}$

lalu eliminasi persamaan 1 dan 2 untuk mencari a dan b.

(caranya Terlampir) *

maka nilai a = 2 dan b = -5

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Nomor 3

Diketahui f(x) = 5x + 2 dan g(x) = 3 - 2x.Jika (f o g)(a) = -3, maka nilai a adalah ....

Jawaban :

berarti

$\mathbf{f(a)=5a+2}$
$\mathbf{g(a)=3-2a}$

$\mathbf{(f \circ g)(a)=-3}$
$\mathbf{f\left(g\left(a\right)\right)=-3}$
$\mathbf{f\left(3-2a\right)=-3}$
$\mathbf{5\left(3-2a\right)+2=-3}$
$\mathbf{15-10a+2=-3}$
$\mathbf{17+3=10a}$
$\boxed{\mathbf{a=2}}$