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