( ) x y z. 3 Functions 36. SECTION D Composite Functions

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1 3 Functions 36 SECTION D Composite Functions By the end o this section you will be able to understand what is meant by a composite unction ind composition o unctions combine unctions by addition, subtraction, multiplication and division understand what is meant by equality o unctions D1 Deinition o Composite Function What does the word composite mean in everyday lanuae? Somethin which is made up o separate parts. A composite unction is a unction which is made up o several unctions. A composite unction is also called a unction o a unction. Consider unctions : A B and : B C illustrated below: x ( x ) ( ( x )) A Fi 6 Let x be an element in set A. Then the imae o x under the unction ( x) B C iven by is in set B. Note that set B is the domain (start) o the unction. The imae o ( x) under the unction is iven by ( ( x )) and it is in the set C. The unction rom set A to set C which assins each element x in A the element x in C is called the composition o the unctions and. This is normally ( ) denoted by and verbally stated as composite. Hence or all x in the domain A we have x = x (3.6) This is the unction. Note that we carry out the operation irst and then apply the unction to this result. Example 3 Let : A B and : B C be deined by the ollowin diaram: a b c x y z r s t A B C Fi 7 Compute ( )( a), ( )( b) and ( )( c).

2 3 Functions 37 By applyin (3.6) ( )( x) = ( x) and ollow in the arrows in Fi 7 we have a = a = = Because = ( )( b) = ( ( b) ) [ By (3.6)] = = Because = ( )( c) = ( ( c) ) [ By (3.6)] Because y t a y x r b x = y = t c = y Example 4 Let : and : be deined by x = x and x = x+ i. Determine ( )( 3 ) 3 1 ii. Determine iii. Find the unction ( )( x ) and ( )( x) in each case. What do you notice about your results? i. What does Means evaluate ( 3) ( )( 3) mean? statin the domain and codomain irst and then take o this result. This is add 1 to the arument 3 irst and then square your result. By (3.6) ( )( x) = ( ( x) ) we have ( )( 3) = ( ( 3) ) = ( 3+ 1 ) Because ( x) = x+ 1 = ( 4) = 4 = 16 Because ( x) = x ii. Similarly we have 3 = 3 Deinition o ( ) [ ] ( 3 ) Because = x = x = 9 = = 10 Because x = x+ 1 iii. What is ( )( x) equal to? Aain by (3.6) ( )( x) = x we have ( )

3 ( )( x) = ( x) = x+ 1 Because x = x+ 1 = ( x+ 1) = x + x+ 1 Because ( x) = x Similarly we have x = x Deinition o [ ] Because 3 Functions 38 = x x = x = x + 1 Because ( x) = x+ 1 The domain and codomain or both ( )( x) and ( )( x) is. Notice that ( )( 3) = 16 and ( )( 3) = 10 Also ( )( x) = x + x+ 1 and ( )( x) = x + 1 Hence we conclude that [ Not Equal] The above result is important so we ive it a reerence number: (3.7) [ Not Equal] where and are unctions such that the rane (arriva l) o is in the domain (start) o. Fi 8 The shaded area in iure 8 is the rane o and the domain o. For composite unctions such as care must be taken that the rane o is contained in the domain o. A real unction is a unction whose domain and codomain are real numbers. The unctions and deined in Example 4 are real unctions but the unctions deined in Example 3 are not unless the symbols a, b, c, x, represent real numbers. Example 5 Let the unctions and : be deined by ( x) = 1 x and ( x) = x What is larest domain o so that is a real unction? Computin

4 ( )( x) = ( ( x) ) [ Deinition o ] = 1 x Because x = 1 x = 1 x Because x = x 3 Functions 39 For what values o x is 1 x real? Clearly or 1 x 1 and that is all the real numbers between 1 to 1 inclusive. This is the larest domain o or to be a real unction. We can write this as the set A= { x x and 1 x 1 } or as the interval [ 1, 1]. What is a domain and codomain o the unction? We could have the domain and codomain o the unction as ollows: : A where the set A is the larest domain so that is a real unction. Note that the rane (arrival) o must be in the domain ( start) o or the composite unction. In the above example the codomain o is the set o all the real numbers,, but the rane is only the real numbers between 1 to 1 inclusive. This can be illustrated as: The rane o Fi 9 What ha ppens i the rane o is not in the domain o? Then some elements will not b e deined by the composi te unction. For example, let : be deined by ( x) = x 5 and : + be deined by ( x) = x What is the value o ( )( )? ( )( ) = ( ( ) ) But 3 is not in the domain o numb s, = 3 Because = 5 = 3 because the domain o is all the positive real 3 does not exist or is not deined. + er. Hence The domain o The rane o Fi 30 The shaded part in Fi 30 is the rane o which is not in the domain o. Hence the elements in the shaded part will never be transormed by the unction.

5 3 Functions 40 D Examples o Composition o Functions In this subsection we compute the composition o unctions. Example 6 Let the unctions : and : be deined by the ormulae Find a ormula or (a) (b) (a) Computin : x = x ( ) sin ( x) = and = sin x x x x [ Deinition o ] Because ( x) sin ( x) ( sin ( x) ) Because ( x) x = = = = = sin ( x) Remember ( sin ( x) ) = sin ( x) (b) Computin : ( )( x) = ( ( x) ) [ Deinition o ] ( ) Because ( x ) ( x) ( x) = x x = x = sin Because = sin O course we can have composition o more than unctions as the next example shows. Example 7 Let the unction : be deined by the ormula ( x) = sin ( x) Find a ormula or. This is not diicult but the notation with all the brackets maybe conusin. ( )( x) = ( ) ( x) [ Deinition o ] = ( ) sin ( x) Because ( x) = sin ( x) = ( )( sin ( x) ) = ( )( sin sin ( x) ) [ Because = sin] ( ) ( ) ( ) sin sin ( x ) ( ) sin sin sin ( x ) sin sin sin sin ( x) = = = It is very easy to et conused with all the brackets that is why we used a combination o brackets. The composition o unctions is not the multiplication o unctions but is the unction o a unction. In next section we describe the multiplication, division etc o unctions.

6 3 Functions 41 D3 Other Combination o Functions We can deine addition, subtraction, multiplication and division o unctions as ollows: ± x = x ± x (3.8) (3.9) ( x) = ( x) ( x) ( x) (3.10) ( x) The domain o The domain o is Example 7 Let the unctions Determine a ormula or (i) = provided x 0 x ± and is the set { Domain o } { Domain o } { } { in o } { Domain o } x ( x) Doma 0 : and : be deined by the ormula x = x 1 and x = 3x+ 5 + (ii) (iii) 5 (iv) (v) (i) By (3.8) we have ( x) ( x) ( x ) ( x + = + x = + = ) = 4x + 4 (ii) Applyin (3.8) aain we have = ( )( x) = ( x) ( x) = ( x 1) ( 3x+ 5) = x 6 (iii) Usin (3.8) with a constant unction, 5, taken aw ay rom : 5= ( x) 5 = ( 3x + 5) 5 = 3x (iv) By (3.9) we have = ( )( x) = ( x) ( x) = ( x 1)( 3x+ 5) = 3x + x 5 Expandin Brackets (v) By (3.10) we have [ ]

7 3 Functions 4 = ( x) x x 1 = provided 3x x + 5 The domain in each case is all the real numbers,, apart rom (v). What is the domain in this case? 5 Is all the real numbers excludin where 3x + 5= 0 that is x =. 3 D4 Equal Functions What are equal unctions? Functions and are equal i and only i both unctions, and, have the same domain and also or every x in the domain we have (3.11) ( x) = ( x) This is normally written as =. Example: Let : and : be iven by ( x) x 1 = + and ( y) = y+ 1 Then unctions and are equal, =. Remember the symbols x and y are dummy variab les and just used to deine a ormula or a unction. Example: Let : and : be iven by ( x) = x+ 1 and ( x) = x+ 1 The ormulae or the unctions and are identitical but the domains are dierent, and respectively. These unctio ns are not equal, that is. For unctions to be equal the domains need to be the same. Example 6 Let the unctions : A B and : A B where A = { 1, } and B = { 3, 4} be dein ed by the ormulae x = x+ and x = x x+ Show that =. 4 What is the domain o the iven unctions and? Domain is the set A = { 1, }. That is the domain o and are the two inteers 1 and. Substitutin x = 1 and x = into the iven ormulae or unctions: ( x) = x+ and ( x) = x x+ 4 yields 1 = 1+ = 3 and 1 = = 3 () = + = = ( ) 4 and + 4 = 4

8 3 Functions 43 Since ( x) = ( x) or all x values in the domain we conclude =. SUMMARY The composition o unctions is a unction o a unction and is denoted by the symbol and deined by x = x Remember the order is important. For ( )( x) apply the unction to this result. The rane o must be in the domain o Other combinations are (3.8) ± x = x ± x (3.9) ( x) = ( x) ( x) ( x) (3.10) ( x) we evaluate. Also = provided x 0 x [ Not Equal] x irst and then Functions and are equal, =, i and only i or all x in the domain ( x) = ( x)