# 2.3.b operations on functions

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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

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 xs 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 21 2 xxx23

3 xx

xgxf

g

fx.d

22

1

xx

x

12

1

x,x

211

xx

x

OPERATIONS ON FUNCTIONS

Illustration. If and

, then

1 xxf 22 xxxg

xgfxgf.e

22 xxf

12 xx

122 xx

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

xgxfxgf.a

xx 2

xgxfxgf.b

xx 2

OPERATIONS ON FUNCTIONS

Illustration. If and

, then

xxf xxg 2

xgxfxfg.c xx 22

2 xx

xgxf

g

fx.d

x

x

2 x

xx

2

22

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

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

211 2 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

33 25 xxgxxf. ;

gf,g

fand

TIME TO THINK

,xxf. If6

.hgf find ,xxh 3and

,x

xxg

1

,xxgxxgf. andIf 217 .xffind