ACTIVITY 28:

Combining Functions (Section 3.6, pp. 269-275)

The Algebra of Functions:

Let f and g be functions with domains A and B. We define new functions f + g, f − g, fg, and f/g as follows:

BA in Doma)()()( xgxfxgf

BA n Domai)()()( xgxfxgf

BA omain D)()()( xgxfxfg

0g(x)|BAx ain Dom )(

)()(/

xg

xfxgf

Example 1:

Let us consider the functions f(x) = x2 − 2x and g(x) = 3x − 1.Find f + g, f − g, fg, and f/g and their domains:

)(xgf )()( xgxf )13()2( 2 xxx 12 xx

)(xgf )()( xgxf )13()2( 2 xxx 1322 xxx152 xx

)()( xgxf )(xfg )13)(2( 2 xxx xxxx 263 223 xxx 273 23

is for DomainThe f is for DomainThe g

is functions above for the domain thely,Consequent

)(

)(

xg

xf )(/ xgf13

22

x

xx By the above the domain for f/g is all real numbers except when g(x) = 0.

013 x13 x

3

1x

,

3

1

3

1,

Consequently, the domain for f/g is:

Example 2:

Let us consider the functions :

Find f + g, f − g, fg, and f/g and their domains:

1)( and 9)( 22 xxgxxf

)(xgf )()( xgxf )(xgf )()( xgxf

)()( xgxf )(xfg

19 22 xx

19 22 xx

19 22 xx 19 22 xx

that such sx' theall is for DomainThe f 09 2 x 033 xx

33 ]3,3[

that such sx' theall is for DomainThe g 012 x 011 xx

,11,11

3,3A

,11,B 11

33

3,11,3 BA

)(/ xgf)(

)(

xg

xf

1)( and 9)( 22 xxgxxf

1

92

2

x

x

012 xNotice that

12 x1x 1

3,11,3 Consequently, the domain for f/g is

Example 3:

Use graphical addition to sketch the graph of f + g.

Composition of Functions:

Given any two functions f and g, we start with a number x in the domain of g and find its image g(x). If this number g(x) is in the domain of f, we can then calculate the value of f(g(x)). The result is a new function h(x) = f(g(x))obtained by substituting g into f. It is called the composition (or composite) of f and g and is denoted by f ◦ g (read: ‘f composed with g’ or ‘f circle g’)That is we define:

(f ◦ g)(x) = f(g(x)).

Example 4:

Use f(x) = 3x − 5 and g(x) = 2 − x2 to evaluate:

))0((gf ))0(2( 2f )2(f 5)2(3 56 1))4(( ff )5)4(3( f )512( f )7(f 5)7(3 521 16))(( xgf ))(( xgf )2( 2xf 5)2(3 2 x 536 2 x 231 x))0(( fg )5)0(3( g )5(g 2)5(2 252 23)2)(( gg ))2((gg )22( 2g

))(( xfg ))(( xfg )53( xg 2)53(2 x

)42( g )2(g 2)2(2 42 2

)2515159(2 2 xxx )25309(2 2 xx

253092 2 xx 23309 2 xx

Example 5:

Find the functions f ◦ g, g ◦ f, and f ◦ f and their domains.

1)(

x

xxf 12)( xxg

))(( xgf ))(( xgf )12( xf1)12(

12

x

x

x

x

2

12

))(( xfg ))(( xfg

1x

xg 1

12

x

x

11

2

x

x

1

11

1

2

x

x

x

x

1

)1(2

x

xx

1

12

x

xx

1

1

x

x ,11,

,00,

1)(

x

xxf

))(( xff ))(( xff

1x

xf

11

1

xxx

x

11

11

1

xx

xx

xx

11

1

xxx

xx

112

1

xx

xx

12

1

1

x

x

x

x

12

x

x

,

2

1

2

1,11,

Can’t have -1/2 in the domain

But we also can’t have -1 in the domain

Example 6:

Express the function

in the form F(x) = f(g(x)).

4)(

2

2

x

xxF

4)(

x

xxf 2)( xxg

))(()( xgfxF )( 2xf42

2

x

x

Example 7:

Find functions f and g so that f ◦ g = H if

3 2)( xxH

3 2)( xxf xxg )(

))(()( xgfxH ))(( xgf )( xf 3 2 x