operations on functions. definition. sum, difference, product, quotient and composite functions let...
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OPERATIONS ON FUNCTIONS
OPERATIONS ON FUNCTIONS
Definition. Sum, Difference, Product, Quotient and Composite Functions
Let f and g be functions of the variable x.
1. The sum function f+g is the function defined by
xgxfxgf
OPERATIONS ON FUNCTIONS
Let f and g be functions of the variable x.
2. The difference function f – g is the function defined by
xgxfxgf
OPERATIONS ON FUNCTIONS
Let f and g be functions of the variable x.
3. The product function fg is the function defined by
xgxfxfg
OPERATIONS ON FUNCTIONS
Let f and g be functions of the variable x.
4. The quotient function is the function defined by
0
xg,
xg
xfx
g
f
fg–
OPERATIONS ON FUNCTIONS
Let f and g be functions of the variable x.
5. The composite function f g is the function defined by
xgfxgf
O
OPERATIONS ON FUNCTIONS
The domain of f+g , f – g, fg, and
is the intersection of the domains of f and g, with the added restriction on the quotient.
The domain of consists of the set of x’s for which g(x) is contained in the domain of f.
fg–
Of g
OPERATIONS ON FUNCTIONS
Illustration. If and
, then
1xxf
22 xxxg
xgxfxgf.a 1 x 22 xx 12 x
xgxfxgf.b 1 x 22 xx
322 xx
OPERATIONS ON FUNCTIONS
Illustration. If and
, then
1xxf
22 xxxg
xgxfxfg.c 21 2 xxx
233 xx
xgxf
gf
x.d
221
xx
x
12
1
x,x
211
xx
x
OPERATIONS ON FUNCTIONS
Illustration. If and
, then
1xxf
22 xxxg
xgfxgf.e
22 xxf
12 xx
122 xx
OPERATIONS ON FUNCTIONS
Illustration. Given and
.
1xxf
22 xxxg
Rf Dom Rg Domand
gfgf DomDomDom RRR
fg,f-g DomDom also iswhich
OPERATIONS ON FUNCTIONS
Illustration. Given and
.
1xxf
22 xxxg
Rf Dom Rg Domand
0
xggf
gf
andDomDomDom
21,xR and
21,R
OPERATIONS ON FUNCTIONS
Illustration. Given and
.
1xxf
22 xxxg
Rf Dom Rg Domand
fxggxxgf DomandDomDom
R
OPERATIONS ON FUNCTIONS
Illustration. If and
, then
xxf xxg 2
xgxfxgf.a
xx 2
xgxfxgf.b
xx 2
OPERATIONS ON FUNCTIONS
Illustration. If and
, then
xxf xxg 2
xgxfxfg.c xx 222 xx
xgxf
gf
x.d
x
x
2 x
xx
2
2 2
OPERATIONS ON FUNCTIONS
Illustration. If and
, then
xxf xxg 2
xgfxgf.e
xf 2
x 2
4 2 x
OPERATIONS ON FUNCTIONS
Illustration. If and
, then
xxf xxg 2
xfgxfg.f
xg
x 2
OPERATIONS ON FUNCTIONS
Illustration. Given and
.
xxf xxg 2
0 xxfDom 2 xxgDomand
gfgf DomDomDom 20,
fg,f-g DomDom also iswhich
OPERATIONS ON FUNCTIONS
Illustration. Given and
.
xxf xxg 2
0 xxfDom 2 xxgDomand
0
xggf
gf
andDomDomDom
220 x, and
20,
OPERATIONS ON FUNCTIONS
fxggxxgf DomandDomDom
022 xxx and
Illustration. Given and
.
xxf xxg 2
0 xxfDom 2 xxgDomand
2 xx
OPERATIONS ON FUNCTIONS
gxffxxfg DomandDomDom
20 xxx and
Illustration. Given and
.
xxf xxg 2
0 xxfDom 2 xxgDomand
40, 40 xxx and
For numbers 1-5, find
and their respective domains.
TIME TO THINK
211 2 xxgxxf. ;
,fg,gf,gf
121
2
xxgx
xxf. ;
gf,gf and
xxxgx
xf. 41
3 2 ;
For numbers 1-5, find
and their respective domains.
TIME TO THINK
444 xxgxxf. ;
,fg,gf,gf
33 25 xxgxxf. ;
gf,gf and
TIME TO THINK ,xxf. If6
.hgf find ,xxh 3and
,x
xxg
1
,xxgxxgf. andIf 217
.xffind
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