FUNCTIONS
OPERATIONS ON FUNCTIONS
OBJECTIVES:• perform operations on functions;• determine the domain of the given functions;• determine the domain of the resulting functions after performing the operations on functions; and• define a composite function and give its domain.
Definition 0.2.1 (p. 16)
OPERATIONS ON FUNTIONS AND DETERMINATION OF DOMAIN OF FUNCTIONS AS A RESULT OF OPERATION
EXAMPLE 1: (example 1 on p.16)
fandgffggfgffunctionstheforformulasanddomainstheFind
xxgxxfLet
7 , , , ,
3)( and 21)(
Solution:
277)(7)(7
321
)()()(
321)()()(
22321)()()(
:
xxfxf
xx
xgxfx
gf
xxxgxfxfg
xxxxxgxfxgf
areformulasThe
,2,: ,2: gf DD
, , fgandgfgfofdomainsThe
3 except ,2 gf of domain The
fofdomaintheassametheisfofdomainThe 7
EXAMPLE :
x1g(x) ,
x1xf(x) .2
1-xg(x) ,1-x2f(x) 1.
functions. theof domains thestate and
,gf and fg, g,-f g,ffor formulas theFind
2
Definition 0.2.2 (p. 17)
OPERATION ON COMPOSITE FUNCTION AND DETERMINATION OF ITS DOMAIN
EXAMPLE 1: (example 3 on p.17)
domain their and )x(fg and )x(gf Find x)x(g and 3x)x(f Let 2
Solution: 33)()( 2 xxxgfgfforformulaThe
0x ,3x)x)(gf( Therefore,0 is gf of domain the ,thus ;, in lies x)x(g that such ,0
in x all of consists gf of domain the ,,:D the and ,0:D Since fg
3)( 2 xxfgfgforformulaThe
3x of domain natural the is this since,- is domain
the that indicate to need no is there ,3x)x)(fg( Therefore
.,- is fg of domain the ,thus ;0, in lies 3x)x(f that such ,-
in x all of consists fg of domain the ,,0:D the and ,:D Since
2
2
2
gf
EXAMPLE 2: (example 4 on p.18)
)x(hgf Find xh(x) and x1)x(g ,x)x(f Let 3
EXAMPLE 3:
1x1 )d
3x-4 )c
)sin(x )b
1)(x )a
.functions two of ncompositio a as functions following theExpress
3
2
EXAMPLE :
x-1xg(x) ,
11f(x) .2
x-1g(x) , xf(x) 1.
ns.compositio theof domains thestate andgg and f,f f,g g,ffor formulas Find
2
xx
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