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Let a real valued function $f(x)$ has the property $$f(x+2)=\frac{f(x)-5}{f(x)-3}$$ The question is to find out period of $f(x)$

I tried to subsitute $\frac{f(x-2)-5}{f(x-2)-3}$ in place of $f(x)$ in the given equation and ended up getting $$\frac{2f(x-2)-5}{f(x-2)-2}=\frac{f(x)-5}{f(x)-3}$$ I couldnot proceed after this.Any ideas?Thanks.

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    Hint: Take $f(0)=a$, say, and iterate. What are $f(2), f(4), etc. ?$ – 2017-02-21
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    "ended up getting ..."; the very first $2$ in the numerator should not be there. – 2017-02-21
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    @JohnHughes I couldnot get what you mean?I have rechecked and I think there is no problem of $2$ in the numerator – 2017-02-21

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The period is 8.

Steps:

  1. Express f(x+4) in terms of f(x+2) by replacing x with x+2

  2. Using the given equation, express f(x+4) in terms of f(x)

  3. Express f(x+6) in terms of f(x+4)

  4. Using the equation obtained in 2, express f(x+6) in terms of f(x)

  5. Express f(x+8) in terms of f(x+6)

  6. Using the equation obtained in 4, express f(x+8) in terms of f(x), and that should do the trick.

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    Check your grammar and spellings the next time. – 2017-02-21
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    "couldnot" - > "could not" or "couldn't". "Let a real valued function HAVE the property". Anyway, I hope the answer helps. – 2017-02-21