Invers Function

F(x,y)=0

For example:

1. y = 2x -1

y=x other line on the picture

substitute y = x to y = 2x-1

x = 2x-1

x-2x = -1

- x = -1

x = 1…………………………….( i )

Substitute ( i ) to y = x (or y = 2x – 1)

y = 1

Intersection two line y = x and y = 2x – 1 on (1,1)

From y = 2x -1, we will change to become function of x,

y = 2x -1

y – 1 = 2x

½ (y+1) = x

Because y = x and ½ (y+1)

y = ½ (x+1)

We find new function!

y = ½ (x+1)

from all that we calculate above, we get:

f(x) = 2x – 1

g(x) = ½ (x+1)

f( g(x) ) we substitute g(x) to variable on the function f.

g( f(x) ) we substitute f(x) to variable on the function g.

ü f ( g(x) ) = 2 ( g(x) ) - 1

f ( g(x) ) = 2(½ (x+1) - 1

f ( g(x) ) = x + 1 – 1

f ( g(x) ) = x

ü g ( f(x) ) = ½ ((2x-1)+1)

g ( f(x) ) = ½ (2x)

g ( f(x) ) = x

g = f ^-1

f ( g(x) ) = f (f ^-1 (x) ) = x

g( f(x) ) = f (f ^-1 (x) ) = x

2. y=(x-1)/(x+2)

Method 2 find y ^-1

y=(x-1)/(x+2)

y ( x+2 ) = x - 1

xy + 2y = x – 1

xy – x = -1 -2y

x ( y – 1 ) = -1 -2y

x = (-1-2y)/(y-1)

so y ^-1= (-1-2x)/(x-1)