activity 28:
DESCRIPTION
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:. Example 1:. Let us consider the functions f(x) = x 2 − 2x and g(x) = 3x − 1. - PowerPoint PPT PresentationTRANSCRIPT
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