Invertible Functions
It can be verified in general that gof is one-one implies that f is one-one . Similarly , gof is onto implies that g is onto .
Numerical : Let f : { 2 , 3 , 4 , 5 } → { 3 , 4 , 5 , 9 } and g : { 3 , 4 , 5 , 9 } → { 7 , 11 , 15 } be functions defined as f ( 2 ) = 3 , f ( 3 ) = 4 , f ( 4 ) = f ( 5 ) = 5 and g ( 3 ) = g ( 4 ) = 7 and g ( 5 ) = g ( 9 ) = 11 . Find gof .
Solution :
gof ( 2 ) = g ( 3 ) = 7 gof ( 3 ) = g ( 4 ) = 7 gof ( 4 ) = g ( 5 ) = 11 gof ( 5 ) = g ( 5 ) = 11
Invertible Functions
A function f : X → Y is defined to be invertible , if there exists a function g : Y → X such that gof = IX and fog = IY . The function g is called the inverse of f and is denoted by f – 1 .
Thus , if f is invertible , then f must be one-one and onto and conversely , if f is one-one and onto , then f must be invertible
Numerical : Let f : N → Y be a function defined as f ( x ) = 4x + 3 , where , Y = { y ∈ N : y = 4x + 3 for some x ∈ N }. Show that f is invertible . Find the inverse