R – { } and B = R – { }. Consider the function f : A → B defined by f ( x ) = . Is f one-one and onto? Justify your answer.
. Let f : R → R be defined as f ( x ) = x . Choose the correct answer. (A) f is one-one onto (B) f is many-one onto (C) f is one-one but not onto (D) f is neither one-one nor onto.
. Let f : R → R be defined as f ( x ) = x . Choose the correct answer. (A) f is one-one onto (B) f is many-one onto (C) f is one-one but not onto (D) f is neither one-one nor onto.
. Composition of Functions and Invertible Function Definition 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. Fig .
Example Let f : { , , , } → { , , , } and g : { , , , } → { , , } be functions defined as f ( ) = , f ( ) = , f ( ) = f ( ) = and g ( ) = g ( ) = and g ( ) = g ( ) = . Find gof . Solution We have gof ( ) = g ( f ( )) = g ( ) = , gof ( ) = g ( f ( )) = g ( ) = , gof ( ) = g ( f ( )) = g ( ) = and gof ( ) = g ( ) = . Example Find gof and fog , if f : R → R and g : R → R are given by f ( x ) = cos x and g ( x ) = x .