1.9 Composition of Functions

. Composition of Functions When a car driver depresses the accelerator pedal, it controls the flow of fuel which in turn influences the speed of the car. Likewise, the composition of two functions is a kind of ‘chain reaction’, where the functions act upon one after another (Fig. . ). We can explain this further with the concept that a function is a ‘process’. If f and g are two functions then the composition g f x ( ( )) (Fig. . ) is formed in two steps. (i) Feed an input (say x ) to f ; (ii) Feed the output f ( x ) to g to get g f x ( ( )) and call it gf ( x ). Illustration Consider the set A of all students, who appeared in class X of Board Examination. Each student appearing in the Board Examination is assigned a roll number. In order to have confidentiality, the Board arranges to deface the roll number of each student and assigns a code number to each roll number. Let A be the set of all students appearing for the board exam. B Í  be the set all roll numbers and C Í  be the set of all code numbers (Fig. . ). This gives rise to two functions f : ® and g : ® given by b f a = ( ) be the roll number assigned to student a , c g b = ( ) be the code number assigned to roll number b , where a Î , b Î and c Î We can write c g b g f a ( ( )) . Thus, by the combination of these two functions, each student is eventually attached a code number. This idea leads to the following definition. Definition Let f : ® and g : ® be two functions (Fig. . ). Then the composition of f and g denoted by g f  is defined as the function g f x g f x  ( ) ( ( )) ∀ x Î Example . Find f g  and g f  when f x ( ) = and g x ( ) = Solution f x ( ) = , g ( x ) = x – f g x  ( ) = f g x ( ( )) = f x = g f x  ( ) = g f x ( ( )) = g = Thus f g  , g f  . From the above, we see that f g g f   ¹ gf f ( x ) f g g ( f ( x )) Fig. . gf Fig. . Student f ( a ) = b Roll Numbers c = g ( b ) g ( f ( a )) Code Numbers f g gf Fig. . f ( x ) g ( f ( x )) f > g Thinking Corner If f x x m ( ) = and g x x n ( ) = does f g g f   ? Relations and Functions