Discrete Math Notes:
Chapter 5: Functions and Relations
5.4 The inverse of a function
Inverse, is obtained by exchanging the first and second entries in each pair in f
Overview:
Arrow diagram of an inverse function example
X={1,2,3} Y={7,8,9} .7
2
8.
3
9
f:X Y f={(1,7)(2,9).(3,9)}
f-1={7.1).9,2).9,3)}
f -I is not a function. f does not have an inverse
XFX:6 g={(1,9),(2 7).(3,8)}
g-1={7,2).(8,3).(9,1)}
g -1 is a function. g has an inverse defined by g 1(8) = 3 g-1(9)=1
g 1(7) = 2
Captions ^
1. If arrows for f are reversed, the result is not a function because 8 has no outgoing arrow, and also 9 has two outgoing arrows. Thus, f does not have an inverse. 2. If the arrows for g are reversed, the result is a function because each left element has exactly one outgoing arrow. Thus, g has an inverse.
The finite instances show that f-1 is derived by reversing the arrows in the arrow diagram for f. If and only if every element in Y has precisely one outgoing arrow after the arrows are reversed, the resulting f- 1 is a function, which holds if and only if f is a bijection.