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    L2, Composite Functions

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    manobilli30
    Composite functions combine multiple functions performed in sequence. The document provides examples of calculating composite functions fg(x) and gf(x) using given functions f(x) and g(x). It explains that the order of functions matters, as fg(x) and gf(x) may produce different results. The document also shows how to derive a single function that represents the composite of f(x) and g(x) in a given order.

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    © All Rights Reserved
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    L2, Composite Functions

    Uploaded by

    manobilli30
    20 views13 pages
    Composite functions combine multiple functions performed in sequence. The document provides examples of calculating composite functions fg(x) and gf(x) using given functions f(x) and g(x). It explains that the order of functions matters, as fg(x) and gf(x) may produce different results. The document also shows how to derive a single function that represents the composite of f(x) and g(x) in a given order.

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    IGCSE Functions

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    Composite functions combine multiple functions performed in sequence. The document provides examples of calculating composite functions fg(x) and gf(x) using given functions f(x) and g(x). It explains that the order of functions matters, as fg(x) and gf(x) may produce different results. The document also shows how to derive a single function that represents the composite of f(x) and g(x) in a given order.

    Copyright:

    © All Rights Reserved
    Download as PPTX, PDF, TXT or read online from Scribd
    Download as pptx, pdf, or txt
    20 views13 pages

    L2, Composite Functions

    Uploaded by

    manobilli30
    Composite functions combine multiple functions performed in sequence. The document provides examples of calculating composite functions fg(x) and gf(x) using given functions f(x) and g(x). It explains that the order of functions matters, as fg(x) and gf(x) may produce different results. The document also shows how to derive a single function that represents the composite of f(x) and g(x) in a given order.

    Copyright:

    © All Rights Reserved
    Download as PPTX, PDF, TXT or read online from Scribd
    Download as pptx, pdf, or txt
    of 13

    COMPOSITE FUNCTIONS

    Starter
    Answer these questions, based on the following functions:

    𝑓 ( 𝑥 ) =5 𝑥 +5 𝑔 ( 𝑥 )=9 𝑥 − 4 2
    h ( 𝑥 )=𝑥 − 3 𝑥 −4
    1) Find the value of 2) Find the value of x if: 3) Find the value of x if f(x) = 2g(x)
    each function if x = 4 a) f(x) = 65
    b) g(x) = -22
    𝑓 ( 𝑥 ) =5 𝑥 +5 Sub in 4
    𝑓 ( 𝑥 ) =2 𝑔(𝑥) Replace f(x)
    and g(x)
    𝑓 ( 4 )=5( 4)+5 Calculate
    𝑓 ( 𝑥 ) =5 𝑥 +5 Replace f(x) 5 𝑥+5¿ 2(9 𝑥 − 4) Expand
    𝑓 ( 4 )=25 65=5 𝑥+ 5 Subtract 5 bracket
    5 𝑥+5¿ 18 𝑥 − 8 Add 8,
    60=5 𝑥 Divide by 5
    𝑔 ( 𝑥 )=9 𝑥 − 4 Sub in 4 12=𝑥 13=13 𝑥 Subtract 5x
    Divide by 13
    𝑔 ( 4 )=9( 4)− 4 Calculate 1= 𝑥
    𝑔 ( 4 )=32 𝑔 ( 𝑥 )=9 𝑥 − 4 Replace g(x)
    2 −22=9 𝑥 − 4 Add 4
    h ( 𝑥 )=𝑥 − 3 𝑥 −4
    2
    Sub in 4 −18=9 𝑥 Divide by 9
    h ( 4 ) =( 4 ) − 3 ( 4 ) − 4 Calculate − 2= 𝑥
    h ( 4 ) =0
    Composite Functions
    • Today we will be continuing our work on
    functions

    • We will be looking at problems involving


    ‘Composite functions’

    • A Composite Function is one which is


    made up of several functions performed
    in a certain order
    Composite Functions
    Lets start with the following functions:

    𝑓 ( 𝑥 ) =𝑥
    2
    𝑔 ( 𝑥 )=𝑥 − 4

    1a) Calculate fg(8) The function fg(8) means you calculate g(8),
    and then calculate f(the answer)
    𝑔 ( 𝑥 )=𝑥 − 4
    Sub in 8
     So substitute 8 into g(x) to find the
    𝑔 ( 8 )=8 − 4 value of g(8)
    Calculate
    𝑔 ( 8 )=4  Then take the answer, and substitute it
    into f(x)
    2
    𝑓 ( 𝑥 ) =𝑥
    Sub in 4, which was g(8)
    2
    𝑓𝑔 ( 8 )=(4 ) We are calculating ‘f of g of 8’
    Calculate
    𝑓 𝑔 ( 8 )=16

    Therefore: 𝑓𝑔( 8)=16


    Composite Functions
    Lets start with the following functions:

    𝑓 ( 𝑥 ) =𝑥
    2
    𝑔 ( 𝑥 )=𝑥 − 4

    1b) Calculate gf(8) The function gf(8) means you calculate f(8),
    and then calculate g(the answer)
    2
    𝑓 ( 𝑥 ) =𝑥
    Sub in 8
    2  So substitute 8 into f(x) to find the
    𝑓 ( 8 ) =( 8) value of f(8)
    Calculate
    𝑓 ( 8 ) =64  Then take the answer, and substitute it
    into g(x)
    𝑔 ( 𝑥 )=𝑥 − 4
    Sub in 64, which was f(8)
    𝑔 𝑓 ( 8 )=64 − 4 We are calculating ‘g of f of 8’
    Calculate
    𝑔 𝑓 ( 8 )=60 Notice that the order is
    important – you get a different
    Therefore:𝑔𝑓 (8)=60 answer if it is changed!
    Composite Functions
    Lets start with the following functions:

    𝑓 ( 𝑥 ) =𝑥
    2
    𝑔 ( 𝑥 )=𝑥 − 4

    1c) Calculate fg(x) The function fg(x) means you calculate g(x),
    and then calculate f(the answer)
    𝑔 ( 𝑥 )=𝑥 − 4 Sub in x (actually,
    nothing happens!)  So substitute x into g(x) to find the
    𝑔 ( 𝑥 )=𝑥 − 4 value of g(x)
     Then take the answer, and substitute it
    𝑓 ( 𝑥 ) =𝑥
    2 into f(x)
    Sub in (x – 4) for all ‘x’
    2 terms (This was g(x))
    𝑓 ( 𝑥 − 4 ) =( 𝑥 −4 )
    2
    Calculate We are calculating ‘f of g of x’
    𝑓 ( 𝑥 − 4 ) =𝑥 −8 𝑥 +16

    2
    Therefore: 𝑓𝑔( 𝑥 )=( 𝑥 − 4)
    Composite Functions
    Lets start with the following functions:

    𝑓 ( 𝑥 ) =3 𝑥 −1 𝑔 ( 𝑥 )=𝑥 2 +1 What this does is give us a


    single function that can replace
    the functions f and g (in this
    particular order)
    2a) Calculate fg(x)
     3x2 + 2
    2
    𝑔 ( 𝑥 )=𝑥 +1
    Replace g(x) with its expression in fg(x) This function is equivalent to
    𝑓𝑔 ( 𝑥¿) 𝑓 ( 𝑥 2 +1) performing x2 + 1 followed by
    3x - 1

    𝑓 ( 𝑥 ) =3 𝑥 −1 So we need to find f(x2 + 1)


     Replace all ‘x’ terms with ‘x2 + 1’ terms
    𝑓 ( 𝑥 2+ 1 )=3( 𝑥 2+ 1) −1
    Expand bracket
    𝑓 ( 𝑥 + 1 )=3 𝑥 +3 −1
    2 2

    Simplify
    𝑓 ( 𝑥 + 1 )=3 𝑥 +2
    2 2
    Composite Functions
    Lets start with the following functions:

    𝑓 ( 𝑥 ) =3 𝑥 −1 𝑔 ( 𝑥 )=𝑥 2 +1 What this does is give us a


    single function that can replace
    the functions f and g (in this
    particular order)
    2a) Calculate gf(x)
     9x2 - 6x + 2
    𝑓 ( 𝑥 ) =3 𝑥 −1
    Replace g(x) with its expression in fg(x) This function is equivalent to
    𝑔𝑓 ( 𝑥¿)𝑔 (3 𝑥 − 1) performing 3x – 1 followed by
    x2 + 1

    2
    𝑔 ( 𝑥 )=𝑥 +1 So we need to find g(3x - 1)
     Replace all ‘x’ terms with ‘3x - 1’ terms
    𝑔 ( 3 𝑥 − 1 )=( 3 𝑥 − 1 )2 +1
    Expand bracket
    2
    𝑔 ( 3 𝑥 − 1 )=9 𝑥 − 6 𝑥+ 1+ 1
    Simplify
    2
    𝑔 ( 3 𝑥 − 1 )=9 𝑥 − 6 𝑥+ 2
    Plenary
    2
    𝑓 ( 𝑥 ) =𝑥 +8 𝑔 ( 𝑥 )=𝑥 + 4
    Set equal to
    Find the value of for which fg(x) = gf(x)
    each other
    𝑓𝑔 ( 𝑥 ) Find fg(x) 𝑔𝑓 ( 𝑥 ) Find gf(x) 𝑓𝑔 ( 𝑥 ) =𝑔𝑓 (𝑥)
    ¿ 𝑓 ( 𝑥 + 4) 2
    ¿ 𝑔 ( 𝑥 + 8) 2 2
    𝑥 + 8 𝑥+24 =¿𝑥 +12
    2
    𝑓 ( 𝑥 ) =𝑥 +8 𝑔 ( 𝑥 )=𝑥 + 4 8 𝑥 +24=¿12
    𝑓 ( 𝑥 + 4 )=( 𝑥+ 4 )2 +8 𝑔 ( 𝑥 2 +8 ) =( 𝑥 2 +8)+ 4 8 𝑥=− 12
    2
    𝑓 ( 𝑥 + 4 )=𝑥 + 8 𝑥+16 +8 𝑔 ( 𝑥 2 +8 ) = 𝑥2 +12 𝑥=−1.5
    2
    𝑓 ( 𝑥 + 4 )=𝑥 + 8 𝑥+ 24
    2
    𝑔𝑓 ( 𝑥 )=𝑥 +12
    2
    𝑓𝑔 ( 𝑥 ) =𝑥 + 8 𝑥 +24
    Prep
    • Exercise 1 page 163-164
    Summary
    • We have looked at composite functions

    • We have seen how they can combine multiple functions into a


    single one

    • Sometimes the combined equation is simpler, and sometimes


    it can be more complicated

    • Composite functions can be used in games to describe motion


     The two separate functions could be based on
    horizontal and vertical movement (eg – throwing a ball)
     You can choose these and then combine them to form
    an equation for the exact location of a ball at a particular time…

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