A function \(f\) from the integers to the integers is defined as follows:

Suppose \(k\) is odd and \(f(f(f(k))) = 27.\) Find \(k\).

Do we have to work backwards?

mathtoo Dec 27, 2018

#1**+2 **

Mmmm....I haven't seen one like this.....but.....here's my best shot....

Let us suppose that the function is

f(k) = 3k and let k = 1

So

f(1) = 3 (1) = 3

f ( f(1) ) = f(3) = 3(3) = 9

f ( f ( f ( 1 ) ) ) = f(9) = 3(9) = 27

So..... k = 1

Anyone else have other thoughts???

CPhill Dec 27, 2018

#2**+1 **

Hmm, this looks very tricky.

Since \(k\) is odd, we denote \(f(k)=k+3.\) Any odd number plus \(3\) is even, so \(f(k + 3) = \frac{k + 3}{2}\).

But, if \(\frac{k + 3}{2}\) is odd, therefore \(f \left( \frac{k + 3}{2} \right) = \frac{k + 3}{2} + 3 = 27.\) Then this leads for \(k=45\), and \(f(f(f(45))) = f(f(48)) = f(24) = 12\), so it is even. \(f \left( \frac{k + 3}{2} \right) = \frac{k + 3}{4} = 27\) leads to \(105\), so this is your answer?

Check: \(f(f(f(105))) = f(f(108)) = f(54) = 27.\)

.neworleans06 Dec 27, 2018

#3**0 **

Wondering if f(x) = x then f(27) = 27

f(f(27) ) = 27

f(f(f(27))) = 27 then k = 27 (which is odd)...... Hmmmmm

ElectricPavlov Dec 27, 2018