Inverse Functions part I

F(x,y)=0
Function y = f(x) : Vertical Line
1.1. Function x = g(y) : Horizontal Line
For example:
1. y= x^2
domain : (y) subset riil number
range : (y) always on positive value
x=g (y) : Horizontal Line : Invertible

2. y = 2x - 1
If we subs zero to x, so y = -1
If we subs zero to y, so x = 1/2
Function y = 2x – 1 is a line which (0, -1) and (1/2, 0) is intersecting graph on axis
y = x other line on the picture
- substitute y = x to y = 2x-1
x =2x-1 making a move internode (from right to left)
x-2x = -1
- x = -1 multiple with -1
x = 1 …………………………….( i )
Substitute ( i / x = 1) 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, so

y = 2x -1 making a move internode (from right to left)
y – 1 = 2x multiple with ½
½ y + ½ = x
½(y + 1) = x
Because y = x and x = ½(x +1)
y = ½(x +1)
We find new function!

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.
 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

 g ( f(x) ) = ½( (2x – 1) +1)
g ( f(x) ) = ½(2x )
g ( f(x) ) = x
g = inverse f
f ( g(x) ) = f (f inverse (x) ) = x
g( f(x) ) = f (f inverse (x) ) = x
3. y = (x - 1) / (x + 2)

Method 2 find y inverse
y = (x - 1) / (x + 2) multiple with 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 inverse = (-1 – 2y) / (y - 1)

When x = 0 y = -1
When y = 0 -1 – 2x = 0
-2x = 1
x = ½