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