Soal-Soal Matematika/Fungsi invers

Bentuk dan sifat fungsi invers

bentuk f^-1(x)

sifat:

1. y = f(x) → f^-1(y) = x
2. f(x-a) = g(x) → f(x) = g(x+a)
3. (f^-1(x))^-1(x) = f(x)
4. f(f^-1(x)) = f^-1(f(x))
5. (f o f^-1) (x) = x
6. (f o g)^-1 (x) = (g^-1 o f^-1) (x)
7. (f o g) (x) = h(x) → (f^-1 o h) (x) = g(x)

tabel fungsi invers

Fungsi invers (f(x))	Hasil (f-1(x))
$ax+b$	${\frac {x-b}{a}}$
$ax^{2}+bx+c$
${\frac {ax+b}{cx+d}}$	${\frac {-dx+b}{cx-a}}$
${(ax+b)}^{2}$	${\frac {{\sqrt {x}}-b}{a}}$
${\sqrt {ax+b}}$	${\frac {x^{2}-b}{a}}$
$a^{bx}$	$^{a^{b}}logx$
$^{a}logbx$	${\frac {a^{x}}{b}}$

contoh soal

1 tentukan fungsi invers dari $f(x)={\frac {2x-1}{3x+4}}$!

Jawaban

$${\begin{aligned}f(x)&={\frac {2x-1}{3x+4}}\\3xf(x)+4f(x)&=2x-1\\4f(x)+1&=2x-3xf(x)\\4f(x)+1&=(2-3f(x))x\\x&={\frac {4f(x)+1}{2-3f(x)}}\\f^{-1}(x)&={\frac {4x+1}{-3x+2}}\\f^{-1}(x)&={\frac {4x+1}{-(3x-2)}}\\f^{-1}(x)&={\frac {-4x-1}{3x-2}}\\\end{aligned}}$$

2 tentukan nilai f(1) jika $f^{-1}(x)={\frac {2x-1}{5x-7}}$!

cara 1

Jawaban

$${\begin{aligned}{\text{misalkan}}f(1)&=a{\text{maka}}f^{-1}(a)&=1\\{\text{untuk}}a&=x\\f^{-1}(x)&={\frac {2x-1}{5x-7}}\\1&={\frac {2x-1}{5x-7}}\\5x-7&=2x-1\\3x&=6\\x&=2\\a&=x\\a&=2\\\end{aligned}}$$

cara 2

Jawaban

$${\begin{aligned}f^{-1}(x)&={\frac {2x-1}{5x-7}}\\{\text{diubah menjadi}}f({\frac {2x-1}{5x-7}})&=x\\f({\frac {2x-1}{5x-7}})&=x\\{\text{anggap bahwa}}f(1)&=f({\frac {2x-1}{5x-7}})\\1&={\frac {2x-1}{5x-7}}\\5x-7&=2x-1\\3x&=6\\x&=2\\f({\frac {2x-1}{5x-7}})&=x\\f({\frac {2(2)-1}{5(2)-7}})&=2\\f({\frac {3}{3}})&=2\\f(1)&=2\\\end{aligned}}$$

cara 3

Jawaban

$${\begin{aligned}f^{-1}(x)&={\frac {2x-1}{5x-7}}\\{\text{diubah menjadi}}f(x)&={\frac {7x-1}{5x-2}}\\f(x)&={\frac {7x-1}{5x-2}}\\f(x)&={\frac {7(1)-1}{5(1)-2}}\\f(x)&={\frac {6}{3}}\\f(x)&=2\\\end{aligned}}$$

3 tentukan nilai f^-1(1) jika $f({\frac {3}{2x-3}})={\frac {2x+3}{x+4}}$!

cara 1

Jawaban

$${\begin{aligned}{\text{misalkan}}f^{-1}(1)&=a{\text{maka}}f(a)&=1\\{\text{untuk}}a&={\frac {3}{2x-3}}\\f({\frac {3}{2x-3}})&={\frac {2x+3}{x+4}}\\1&={\frac {2x+3}{x+4}}\\x+4&=2x+3\\x&=1\\a&={\frac {3}{2x-3}}\\a&={\frac {3}{2(1)-3}}\\a&=-3\\\end{aligned}}$$

cara 2

Jawaban

$${\begin{aligned}f({\frac {3}{2x-3}})&={\frac {2x+3}{x+4}}\\{\text{diubah menjadi}}f^{-1}({\frac {2x+3}{x+4}})&={\frac {3}{2x-3}}\\f^{-1}({\frac {2x+3}{x+4}})&={\frac {3}{2x-3}}\\{\text{anggap bahwa}}f^{-1}(1)&=f^{-1}({\frac {2x+3}{x+4}})\\1&={\frac {2x+3}{x+4}}\\x+4&=2x+3\\x&=1\\f^{-1}({\frac {2x+3}{x+4}})&={\frac {3}{2x-3}}\\f^{-1}({\frac {2(1)+3}{(1)+4}})&={\frac {3}{2(1)-3}}\\f^{-1}({\frac {5}{5}})&={\frac {3}{-1}}\\f^{-1}(1)&=-3\\\end{aligned}}$$