My Math Forum Generalizing with a function

 Pre-Calculus Pre-Calculus Math Forum

 August 20th, 2017, 11:29 AM #11 Senior Member   Joined: May 2016 From: USA Posts: 1,030 Thanks: 420 Continuing on $f^1(x) = \dfrac{x}{\sqrt{1 + cx^2}} \implies$ $f^2(x) = \dfrac{\dfrac{x}{\sqrt{1 + cx^2}}}{\sqrt{1 + c \left ( \dfrac{x}{\sqrt{1 + cx^2}} \right )^2}} = \dfrac{\dfrac{x}{\sqrt{1 + cx^2}}}{\sqrt{1 + \dfrac{cx^2}{1 + cx^2}}} =$ $\dfrac{\dfrac{x}{\sqrt{1 + cx^2}}}{\sqrt{ \dfrac{1 + cx^2 + cx^2}{1 + cx^2}}} = \dfrac{\dfrac{x}{\cancel {\sqrt{1 + cx^2}}}}{\dfrac{\sqrt{1 + 2cx^2}}{\cancel {\sqrt{1 + cx^2}}}} = \dfrac{x}{\sqrt{1 + 2cx^2}}.$ Now let's try $f^3(x) = \dfrac{f^2(x)}{\sqrt{1 + c * \{f^2(x)\}^2}} = \dfrac{\dfrac{x}{\sqrt{1 + 2cx^2}}}{\sqrt{1 + c * \left \{\dfrac{x}{\sqrt{1 + 2cx^2}} \right \}^2}} =$ $\dfrac{\dfrac{x}{\sqrt{1 + 2cx^2}}}{\sqrt{1 + \dfrac{cx^2}{1 + 2cx^2}}} = \dfrac{\dfrac{x}{\sqrt{1 + 2cx^2}}}{\sqrt{ \dfrac{1 + 2cx^2 + cx^2}{1 + 2cx^2}}} =$ $\dfrac{\dfrac{x}{\cancel{\sqrt{1 + 2cx^2}}}}{\dfrac{\sqrt{1 + 3cx^2}}{\cancel{\sqrt{1 + 2cx^2}}}} = \dfrac{x}{\sqrt{1 + 3cx^2}}.$ I see a pattern namely $f^n(x) = \dfrac{x}{\sqrt{1 + cnx^2}}.$ Now all you have to do is to prove it.
August 20th, 2017, 12:11 PM   #12
Member

Joined: Sep 2014
From: Morocco

Posts: 36
Thanks: 0

Hi Jeff, thanks for the semantic clarification.

Quote:
 Originally Posted by Mifarni14 Hi again, sorry for the late reply. I simplified f(x) and f(f(x)) again, it turns out that I had to simplify more. What I then did was to conjecture the following: For each n in $\displaystyle \mathbb{N^*}$, (n iterations) $\displaystyle f \circ f \circ f \circ ... \circ f(x) = \frac{x}{\sqrt{1+ncx^2}}$ This was later demonstrated through induction.
Thanks for the help nonetheless.

 Tags function, generalizing

 Thread Tools Display Modes Linear Mode

 Similar Threads Thread Thread Starter Forum Replies Last Post vlekje5 Pre-Calculus 11 March 27th, 2017 12:58 PM MisaKr Calculus 2 October 24th, 2016 11:13 AM standardmalpractice Math 1 March 21st, 2016 08:47 AM msgelyn Number Theory 2 January 12th, 2014 03:13 AM boyo Applied Math 1 November 20th, 2009 08:39 AM

 Contact - Home - Forums - Cryptocurrency Forum - Top