q

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
