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Math Help - functions prob

  1. #1
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    functions prob

    salam sahabat.please help me to solve this question.
    if f(x) = ax + b , and f^3(x) = 64x + 21 , find the rule of f^n(x).i appreciate everyone's help.
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  2. #2
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    Hello, muslim!

    \text{If }f(x) \,=\, ax + b,\,\text{ and }f^3(x) \,=\, 64x + 21,\:\text{find the rule of }f^n(x).

    f(x) \:=\:ax+b

    f^2(x)\;=\;f(f(x)) \;=\;f(ax+b)\;=\;a(ax+b) + b \;=\;a^2x + ab + b

    f^3(x)\;=\;f(f(f(x))) \;=\;f(a^2x+ab + b) \;=\;a(a^2x+ab+b) + b \;=\;a^3x + (a^2+a+1)b


    Since f^3(x) \:=\:64x + 21
    . . we have: . a^3x + (a^2+a+1)b \;=\;64x + 21

    Equating coefficients: . \begin{array}{cc}a^3 \;=\;64 & [1] \\ (a^2+a+1)b \;=\;21 & [2] \end{array}

    From [1]: . a^3 \:=\:64\quad\Rightarrow\quad a \,=\,4

    Substitute into [2]: . (4^2+4+1)b \:=\:21 \quad\Rightarrow\quad 21b \:=\:21 \quad\Rightarrow\quad b \:=\:1

    Hence:. . f(x) \:=\:4x + 1


    We have:

    . . \begin{array}{ccccccc}<br />
f(x) &=& & =& 4x + 1 \\<br />
f^2(x) &=& f(4x+1) &=& 16x + 5 \\<br />
f^3(x) &=& f(16x+5) &=& 64x + 21 \\<br />
f^4(x) &=& f(64x+21) &=& 256x + 85 \end{array}


    Therefore: . f^n(x) \;=\;4^nx + \frac{4^n-1}{3}

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  3. #3
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    thanx

    Quote Originally Posted by Soroban View Post
    Hello, muslim!


    f(x) \:=\:ax+b

    f^2(x)\;=\;f(f(x)) \;=\;f(ax+b)\;=\;a(ax+b) + b \;=\;a^2x + ab + b

    f^3(x)\;=\;f(f(f(x))) \;=\;f(a^2x+ab + b) \;=\;a(a^2x+ab+b) + b \;=\;a^3x + (a^2+a+1)b


    Since f^3(x) \:=\:64x + 21
    . . we have: . a^3x + (a^2+a+1)b \;=\;64x + 21

    Equating coefficients: . \begin{array}{cc}a^3 \;=\;64 & [1] \\ (a^2+a+1)b \;=\;21 & [2] \end{array}

    From [1]: . a^3 \:=\:64\quad\Rightarrow\quad a \,=\,4

    Substitute into [2]: . (4^2+4+1)b \:=\:21 \quad\Rightarrow\quad 21b \:=\:21 \quad\Rightarrow\quad b \:=\:1

    Hence:. . f(x) \:=\:4x + 1


    We have:

    . . \begin{array}{ccccccc}<br />
f(x) &=& & =& 4x + 1 \\<br />
f^2(x) &=& f(4x+1) &=& 16x + 5 \\<br />
f^3(x) &=& f(16x+5) &=& 64x + 21 \\<br />
f^4(x) &=& f(64x+21) &=& 256x + 85 \end{array}


    Therefore: . f^n(x) \;=\;4^nx + \frac{4^n-1}{3}

    but sorobon , how you identify [(4^n) - 1 / 3]??can you show me the steps??
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