Tolong bantuin saya !! - Universityku

Pertanyaan :
1. Jika f(x) = $frac{4 - x}{x - 1}, x neq 1$ dan g(x) = $frac{x}{2x+1} ,xneq -frac{1}{2}$ , tentukan (fog⁻¹)⁻¹(x) adalah

2. Tentukan hasil dari $frac{log^25-log 25times log 3+log^23}{log 5-log 3}$ adalah

Tolong bantuin saya !!

Jawaban Terkonfirmasi

Soal no. 1

Jika $f(x)=dfrac{4-x}{x-1}, x neq 1$ dan $g(x)=dfrac{x}{2x+1}, xneq-frac{1}{2}$ , maka:

$boxed{,begin{aligned}&left(fcirc g^{-1}right)^{-1}(x)=frac{x+4}{3x+9}\&qquad{sf dengan} xne-3end{aligned},}$

Soal no. 2

$begin{aligned}&frac{log^25-log25timeslog3+log^23}{log 5-log3}\&= boxed{,log5-log3,}\&= boxed{,logleft(frac{5}{3}right),}end{aligned}$
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Pembahasan

Soal no. 1

Pokok bahasan: Komposisi dan Invers Fungsi

Diketahui
$begin{aligned}f(x)&=frac{4-x}{x-1}, x neq 1\g(x)&=frac{x}{2x+1}, xneq-frac{1}{2}end{aligned}$

Ditanyakan

$left(fcirc g^{-1}right)^{-1}(x)$

Penyelesaian

Pada komposisi fungsi, berlaku:

$(fcirc g)^{-1}(x)=left(g^{-1}circ f^{-1}right)(x)$

Maka,

$begin{aligned}left(fcirc g^{-1}right)^{-1}(x)&=left(left(g^{-1}right)^{-1}circ f^{-1}right)(x)\Rightarrow left(fcirc g^{-1}right)^{-1}(x)&=left(gcirc f^{-1}right)(x)end{aligned}$

Kita tentukan $f^{-1}(x)$ terlebih dahulu.

$begin{aligned}f(x)&=frac{4-x}{x-1}=y\y(x-1)&=4-x\xy-y&=4-x\xy+x&=y+4\x(y+1)&=y+4\x=f(y)&=frac{y+4}{y+1}\Rightarrow f^{-1}(x)&=frac{x+4}{x+1},, xneq-1end{aligned}$

Sehingga,

$begin{aligned}&left(fcirc g^{-1}right)^{-1}(x)\&=left(gcirc f^{-1}right)(x)\&=frac{f^{-1}(x)}{2f^{-1}(x)+1}\&=left(frac{x+4}{x+1}right)Bigg/left(2left(frac{x+4}{x+1}right)+1right)\&=left(frac{x+4}{x+1}right)Bigg/left(frac{2(x+4)+x+1}{x+1}right)\&=left(frac{x+4}{cancel{x+1}}right)Bigg/left(frac{3x+9}{cancel{x+1}}right)\&=boxed{,frac{x+4}{3x+9},, xne-3,}end{aligned}$
$blacksquare$

Soal no. 2

Pokok bahasan: Logaritma

$begin{aligned}&frac{log^25-log25timeslog3+log^23}{log 5-log3}\&{= }frac{log^25-logleft(5^2right)timeslog3+log^23}{log 5-log3}\&{= }frac{overbrace{log^25}^{begin{array}{c}a^2end{array}}:-:overbrace{2log5timeslog3}^{begin{array}{c}2abend{array}}:+:overbrace{log^23}^{begin{array}{c}b^2end{array}}}{log 5-log3}\&quadtoleft[ begin{aligned}a^2-2ab+b^2=(a-b)^2end{aligned} right]\&{= }frac{(log5-log3)^2}{log 5-log3}end{aligned}$
$begin{aligned}&{= }boxed{,log5-log3,}=boxed{,logleft(frac{5}{3}right),}end{aligned}$
$blacksquare$
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Pelajari Lebih Lanjut

Contoh soal tentang Komposisi dan Invers Fungsi

Contoh soal tentang Logaritma

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Detail Jawaban

Soal No. 1
Mata Pelajaran: Matematika
Kelas: 10 (X)
Materi: Fungsi
Kode Kategorisasi: 10.2.3
Soal No. 2
Mata Pelajaran: Matematika
Kelas: 10 (X) – Peminatan
Materi: Fungsi Eksponen dan Fungsi Logaritma
Kode Kategorisasi: 10.2.1