SOLUTION: If f(x) = x/sqr(1+x^2), prove by mathematical induction for n>=2 that f(f(...(f (x))...)) = x/sqr(1+nx^2), if there are n number of letter f's on the LHS. Please help! I don't u

Question 1136739: If f(x) = x/sqr(1+x^2), prove by mathematical induction for n>=2 that f(f(...(f (x))...)) = x/sqr(1+nx^2), if there are n number of letter f's on the LHS.
Please help! I don't understand what f of f(x) means. What exactly am I subbing in when I am trying to prove true for n=2 and assuming true for n=k?
Thank you.
Answer by Theo(13342) (Show Source): You can put this solution on YOUR website!
if you have f(x) = x / sqrt(1 + x^2), then:

f(f(x)) simply replaces x with f(x) in the original equation.

therefore:

is the original equation, and ......

this may be easier to see in my worksheet shown below.

$$$

in my worksheet, i carried it through to the end and got the final answer of f(f(x)) = x / sqrt(1 + 2x^2)

if follows that, if you then do f(f(f(x)),the answer will be x / sqrt(1 + 3x^3), etc.

here's a reference on composite functions which you may already be familiar with.

https://www.mathsisfun.com/sets/functions-composition.html

here's a much simpler example of f(f(x)).

assume f(x) = x^2.

f(f(x)) would then replace x with x^2 and you would get f(f(x)) = (x^2)^2.

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