Section 4.2 Operations with Functions Section 4.2 Operations with Functions.
2.3.b operations on functions
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Transcript of 2.3.b 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
f g –
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.
f g –
O f g
OPERATIONS ON FUNCTIONS
Illustration. If and
, then
1 xxf
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
1 xxf
22 xxxg
xgxfxfg.c 212 xxx
233 xx
xg
xf
g
fx.d
22
1
xx
x
12
1
x,x
211
xx
x
OPERATIONS ON FUNCTIONS
Illustration. Given and
.
1 xxf
22 xxxg
Rf Dom Rg Domand
gfgf DomDomDom
RRR
fg,f-g DomDom also iswhich
OPERATIONS ON FUNCTIONS
Illustration. Given and
.
1 xxf
22 xxxg
Rf Dom Rg Domand
0
xggf
g
fandDomDomDom
21,xR and
21,R
OPERATIONS ON FUNCTIONS
Illustration. Given and
.
1 xxf
22 xxxg
Rf Dom Rg Domand
fxggxxgf DomandDomDom
R
OPERATIONS ON FUNCTIONS
Illustration. If and
, then
xxf
xxg 2
xgxfxfg.c xx 2
22 xx
xg
xf
g
fx.d
x
x
2 x
xx
2
22
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
g
fandDomDomDom
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
2112 xxgxxf. ;
,fg,gf,gf
121
2
xxgx
xxf. ;
gf,g
fand
xxxgx
xf. 41
32 ;
For numbers 1-5, find
and their respective domains.
TIME TO THINK
444 xxgxxf. ;
,fg,gf,gf
3325 xxgxxf. ;
gf,g
fand