+ , x is a real number. (iii) f ( x ) = x , x is a real number. Miscellaneous Examples Example Let R be the set of real numbers. Define the real function f : R → R by f ( x ) = x + and sketch the graph of this function.
Solution Here f ( ) = , f ( ) = , f ( ) = , ..., f ( ) = , etc., and f (– ) = , f (– ) = , ..., f (– ) = and so on. Therefore, shape of the graph of the given function assumes the form as shown in Fig . . Remark The function f defined by f ( x ) = mx + c , x ∈ R , is called linear function , where m and c are constants.
Above function is an example of a linear function . Fig . RELATIONS AND FUNCTIONS Example Let R be a relation from Q to Q defined by R = {( a , b ): a , b ∈ Q and a – b ∈ Z }. Show that ( a , a ) ∈ R for all a ∈ Q ( a , b ) ∈ R implies that ( b , a ) ∈ R (iii) ( a , b ) ∈ R and ( b , c ) ∈ R implies that ( a , c ) ∈ R Solution Since, a – a = ∈ Z , if follows that ( a , a ) ∈ R.
( a , b ) ∈ R implies that a – b ∈ Z . So, b – a ∈ Z . Therefore, ( b , a ) ∈ R (iii) ( a , b ) and ( b , c ) ∈ R implies that a – b ∈ Z . b – c ∈ Z .
So, a – c = ( a – b ) + ( b – c ) ∈ Z . Therefore, ( a , c ) ∈ R