Function: Not onto
Function: one-one and onto( or bijective)
A function f: X → Y is said to be one-one and onto( or bijective), if f is both one-one and onto.
Numerical: Let A be the set of all 50 students of Class X in a school. Let f: A →N be function defined by f( x) = roll number of the student x. Show that f is one-one but not onto.
Solution: Every student in the class has unique roll number, so it is 1-1.
Only roll number 1-50 is assigned to students. Other roll numbers 51 onwards are free & don’ t point to any student, so it is not onto.
Composition of functions
Let f: A → B and g: B → C be two functions. Then the composition of f and g, denoted by gof, is defined as the function gof: A → C given by gof( x) = g( f( x)), ∀ x ∈ A.