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))
$ax+b$	$\frac{x-b}{a}$
$\frac{ax+b}{cx+d}$	$\frac{-dx+b}{cx-a}$
${(ax+b)}^2$	$\frac{\sqrt{x}-b}{a}$
$\sqrt{ax+b}$	$\frac{x^2-b}{a}$
$a^{(bx)}$	$\frac{{}^{a}logx}{b}$
$a^{}logbx$	$\frac{a^x}{b}$

contoh soal

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

Jawaban

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

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

cara 1

Jawaban

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

cara 2

Jawaban

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

cara 3

Jawaban

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

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

cara 1

Jawaban

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

cara 2

Jawaban

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