The document discusses operations on functions and determining the domain of functions after operations. It provides examples of finding formulas and domains for: 1) Adding, subtracting and composing two functions f and g. 2) Determining the domains of composite functions fg and gf given f(x) and g(x). 3) Expressing a composition of two functions as a single composite function. The examples solve for formulas and state domains for added, subtracted, and composed functions. They also find domains for composite functions fg and gf.

1. FUNCTIONS

2. 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.

3. Definition 0.2.1 (p. 16)
OPERATIONS ON FUNTIONS AND DETERMINATION OF DOMAIN OF
FUNCTIONS AS A RESULT OF OPERATION

4. EXAMPLE 1: (example 1 on p.16)
fand
g
f
fggfgffunctionstheforformulasanddomainstheFind
xxgxxfLet
7,,,,
3)(and21)(
−+
−=−+=
Solution:
( ) ( ) ( )
( ) ( )( )
( ) 277)(7)(7
3
21
)(
)(
)(
321)()()(
22321)()()(
:
−+==
−
−+
==





−−+==
−+−=−+−+=+=+
xxfxf
x
x
xg
xf
x
g
f
xxxgxfxfg
xxxxxgxfxgf
areformulasThe
[ ) ( ) [ )∞+=+∞∞−∩∞+ ,2,:,2: gf DD
,, fgandgfgfofdomainsThe −+
[ ) { }3except,2
g
f
ofdomainThe +∞=
fofdomaintheassametheisfofdomainThe 7

5. EXAMPLE :
x
1
g(x),
x1
x
f(x).2
1-xg(x),1-x2f(x)1.
functions.theofdomainsthestateand
,
g
fandfg,g,-fg,fforformulastheFind
2
=
+
=
==
+

6. Definition 0.2.2 (p. 17)
OPERATION ON COMPOSITE FUNCTION AND DETERMINATION OF
ITS DOMAIN

7. EXAMPLE 1: (example 3 on p.17)
( ) ( ) domaintheirand)x(fgand)x(gfFind
x)x(gand3x)x(fLet 2

=+=
Solution:
( ) 33)()( 2
+=+== xxxgfgfforformulaThe 
[ ) ( )
[ ) ( ) [ )
0x,3x)x)(gf(Therefore
,0isgfofdomainthe,thus;,inliesx)x(gthatsuch,0
inxallofconsistsgfofdomainthe,,:Dtheand,0:DSince fg
≥+=
+∞+∞∞−=+∞
+∞∞−+∞



( ) 3)( 2
+== xxfgfgforformulaThe 
( ) [ )
( ) [ )
( )
( ) 3xofdomainnaturaltheisthissince,-isdomain
thethatindicatetoneednoisthere,3x)x)(fg(Therefore
.,-isfg
ofdomainthe,thus;0,inlies3x)x(fthatsuch,-
inxallofconsistsfgofdomainthe,,0:Dtheand,:DSince
2
2
2
gf
++∞∞
+=
+∞∞
+∞+=+∞∞
+∞+∞∞−




8. EXAMPLE 2: (example 4 on p.18)
( ) )x(hgfFind
xh(x)andx1)x(g,x)x(fLet 3

===
EXAMPLE 3:
1x
1
)d
3x-4)c
)sin(x)b
1)(x)a
.functionstwoofncompositioaasfunctionsfollowingtheExpress
3
2
+
+

9. EXAMPLE :
x-1
x
g(x),
1
1
f(x).2
x-1g(x),xf(x)1.
ns.compositiotheofdomainsthestateand
ggandf,ff,gg,fforformulasFind
2
=
−
+
=
==
x
x


10. EXAMPLE :
x-1
x
g(x),
1
1
f(x).2
x-1g(x),xf(x)1.
ns.compositiotheofdomainsthestateand
ggandf,ff,gg,fforformulasFind
2
=
−
+
=
==
x
x
