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Thread: questions on functions

  1. #1
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    questions on functions

    Given that
    f(x)= $\displaystyle \frac{5-x}{1-x}$

    (a) Explain why f has an inverse and show that $\displaystyle f^{-1}=f$.
    (b) Evaluate $\displaystyle f^{51}(4)$.

    i can do part a,
    but have no idea how to start for part b.
    any help is appreciated. =)
    Last edited by wintersoltice; Aug 7th 2011 at 01:05 AM.
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  2. #2
    MHF Contributor Siron's Avatar
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    Re: questions on functions

    I guess you've to calculate the 51th derivative of the function $\displaystyle f$, I should try to recognize a pattern in you derivatives.
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  3. #3
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    Re: questions on functions

    $\displaystyle f(f(x))= \frac{5- \frac{5- x}{1- x}}{1- \frac{5- x}{1- x}}$
    Multiply both numerator and denominator by 1- x:
    $\displaystyle f(f(x))= \frac{5(1- x)- (5- x)}{1- x- (5- x)}= \frac{ 5-5x-5+ x}{1- x-5+ x}= \frac{-4x}{-4}= x$

    That shows both that f is invertible and that $\displaystyle f^{-1}(x)= f(x)$.
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  4. #4
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    Re: questions on functions

    Quote Originally Posted by wintersoltice View Post
    Given that f(x)= $\displaystyle \frac{5-x}{1-x}$
    (a) Explain why f has an inverse and show that $\displaystyle f^{-1}=f$.

    (b) Evaluate $\displaystyle f^{51}(4)$.
    I think that the notation $\displaystyle f^{51}$ refers to function composition.
    For example: $\displaystyle f^5 (x) = f \circ f \circ f \circ f \circ f(x) = f\left( {f\left( {f\left( {f\left( {f(x)} \right)} \right)} \right)} \right) = ?$
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  5. #5
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    Re: questions on functions

    Good point! And the fact that f is its own inverse makes that trivial!
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  6. #6
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    Re: questions on functions

    i found the solution from my school's online portal.

    but i don't understand the steps. i mean from step 2 to 3.

    step1$\displaystyle f^{51} (4)$
    step2$\displaystyle =f[f^{50}(4)]$
    step3$\displaystyle =f(4)$ as $\displaystyle (# f^2(x)=f^{-1}f(x)=x)$
    step4$\displaystyle =\frac{5-4}{1-4}$
    step5$\displaystyle =\frac{-1}{3}$
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  7. #7
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    Re: questions on functions

    Do you understand reply #4?
    Because $\displaystyle f=f^{-1}$ that means $\displaystyle f^2(x)=f\circ f(x)=f(f(x))=x$.
    So $\displaystyle f^5(x)=f(f^4(x))=f(x)$.
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  8. #8
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    Re: questions on functions

    Quote Originally Posted by Siron View Post
    I guess you've to calculate the 51th derivative of the function $\displaystyle f$, I should try to recognize a pattern in you derivatives.
    That would be true if the notation were, instead,

    $\displaystyle f^{(51)}(x),$

    with the parentheses in the exponent.
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