2.3.b operations on functions

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Transcript of 2.3.b operations on functions

  • 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