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Thread: Composite Function

  1. #1
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    Composite Function

    hi!

    Attached is a function qns......now studying for a test and come across problems which i have difficulty completing.... thanks for ur help in advance!
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  2. #2
    Grand Panjandrum
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    A function $\displaystyle f$ is defined by:

    $\displaystyle f:x\mapsto ax+b,\ x \in \mathbb{R}$,

    where $\displaystyle a$ and $\displaystyle b$ are positive constants.

    (a) i. Find $\displaystyle f^3(x),$ (where $\displaystyle f^3(x)=fff(x)$).

    $\displaystyle f^2(x) = f(f(f(x))) = f(f(ax+b))$


    $\displaystyle =f(a(ax+b)+b)$
    $\displaystyle =a(a(ax+b)+b)+b$
    $\displaystyle =a^3x+a^2b+ab+b$
    (ii) Given that $\displaystyle f^3(x)=64x+21$, find the values of $\displaystyle a$ and $\displaystyle b$.

    From part (i) we know that the coefficient of $\displaystyle x$ is $\displaystyle a^3$, so we have $\displaystyle a^3=64$. or $\displaystyle a=4$.

    We also know that the constant term is $\displaystyle a^2b+ab+b=21$, which when we substitute in the known value of $\displaystyle a$ gives us $\displaystyle b=1$.

    RonL

    (other parts to follow in another post if someone else does not provide their solutions first).
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  3. #3
    Grand Panjandrum
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    Quote Originally Posted by CaptainBlack View Post
    A function $\displaystyle f$ is defined by:

    $\displaystyle f:x\mapsto ax+b,\ x \in \mathbb{R}$,

    where $\displaystyle a$ and $\displaystyle b$ are positive constants.
    (iii) With the values of $\displaystyle a$ and $\displaystyle b$ found in part (ii) determin $\displaystyle f^n(x)$, leaving your answer in the form $\displaystyle p^nx+\frac{p^n-1}{q}$ where $\displaystyle p$ and $\displaystyle q$ are integers to be found.

    From the way that part (i) went I will assume that:

    $\displaystyle
    f^n(x)=a^nx + a^{n-1}b + \dots +ab + b
    $

    Putting in $\displaystyle a=4$ and $\displaystyle b=1$:

    $\displaystyle f^n(x)=4^n x + 4^{n+1}+\cdots + 4 + 1$

    Which simplifies to:

    $\displaystyle
    f^n(x)=4^nx + \frac{4^n-1}{3}$


    RonL
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  4. #4
    Grand Panjandrum
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    Another function $\displaystyle g$ is defined by $\displaystyle g: x\mapsto e^x,\ x \in \mathbb{R},\ x<0$.

    (b) (i) Find in terms of $\displaystyle a$ and $\displaystyle b$ the range of $\displaystyle fg$.

    With the definition of $\displaystyle g$ the range of $\displaystyle g$ is the open interval $\displaystyle (0,1)$, which $\displaystyle f$ will map to the open interval $\displaystyle (b, (b+a))$

    (ii) If $\displaystyle h(x)=[g(x)]^2$, determine, with a reason, whether $\displaystyle h^{-1}$ exists.

    With the give devinition of $\displaystyle g(x)$ we have $\displaystyle h(x)=e^{2x}$, and the range of $\displaystyle h$ is the open interval $\displaystyle (0,1)$. But $\displaystyle \ln$ is defined everywhere on this interval and single valued, so if:

    $\displaystyle h(x)=y$,

    then $\displaystyle x=\ln(y)/2$, and is the only such $\displaystyle x$ with image under $\displaystyle h$ of $\displaystyle y$


    RonL
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  5. #5
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    thanks for ur help !
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