Question

"define a mapping \(\gamma:\mathbb{N}\to\mathbb{N}\) such that for each \(n\in\mathbb{N}\) the equation \(\gamma(x)=n\) has exactly three solutions."

suppose i were to change the question to "[...] the equation \(\gamma(x)=n\) has exactly two solutions." then, i would immediately come up with a function\[\gamma(x)=\begin{cases}
(n+1)/2 & \text{ if n is odd,}\\
n/2 & \text{ if n is even}
\end{cases}\]which does this. sadly, i ran out of ideas; i thought about playing with the ranges of definition but failed.

can someone help me? I AM NOT SEEKING FOR A SOLUTION. i want ideas!

Answer 1

ive been at it for three hours

Answer 2

Are polynomials allowed?

Answer 3

anything ^^

Answer 4

Are quadratics allowed?

Answer 5

hmm... are you trying to send me a hint? lol i don't see how i could use polynomials to do this... wait...

Answer 6

One of the forms of quadratics is
y=a(x-x1)(x-x2)
right?

Answer 7

i agree

Answer 8

i got it. that was it :) much obliged, sir!

Answer 9

You're welcome! :)

Answer 10

mathmathe i thought about it a little more and came up with this function\[\gamma(x)=\left \lfloor \frac{1}{3}\left ( x-1 \right ) \right \rfloor\]would it also work?

Answer 11

How many solutions do you suppose it has?

Answer 12

every \(\gamma(x)=n\) is supposed to have 3 solutions.. i think this one does it\[\]

Answer 13

Yes, it should work. I just checked, it's N->N, so there are only three solution!
Good work!

Answer 14

thanks again :)

Answer 15

You're welcome (again)! :)

Answer 16

Yes, I would have recommended this function as well

Answer 17

...or a version of it. Be careful with that function since it can take the value zero, which is not a natural number.

Answer 18

Good thought, James!