March 7th, 2017, 01:31 AM  #1 
Newbie Joined: Sep 2016 From: Maharashtra Posts: 8 Thanks: 0  inverting polynomial
Hi My efforts to find an inverse for this equation: f(x) = (3x^3 + 6x^2 + 7x), exists or not are in vain. Could anyone help me try to solve this. I would be grateful. 
March 7th, 2017, 11:29 AM  #2 
Senior Member Joined: Sep 2015 From: Southern California, USA Posts: 1,481 Thanks: 744 
$f(x) = 3x^3 + 6x^2 + 7x$ $f'(x) = 9x^2 + 12x + 7$ Functions with inverses must be monotonic over their natural or restricted domain. Is $f(x)$ monotonic on $\mathbb{R}$? How might you tell? 
March 7th, 2017, 05:17 PM  #3 
Newbie Joined: Sep 2016 From: Maharashtra Posts: 8 Thanks: 0 
Actually the f(x) is over Zn where n = 9. So the problem is to find if f(x) mod 9 has an inverse or not.

March 7th, 2017, 05:24 PM  #4 
Senior Member Joined: Sep 2015 From: Southern California, USA Posts: 1,481 Thanks: 744  
March 8th, 2017, 01:49 AM  #5 
Newbie Joined: Sep 2016 From: Maharashtra Posts: 8 Thanks: 0 
I am sorry. But could you please help me solving this problem?

March 8th, 2017, 08:06 AM  #6 
Member Joined: Jan 2016 From: Athens, OH Posts: 51 Thanks: 27 
The following table of values of the 9 elements mod 9 shows that the function is indeed both surjective and injective. Hence it does have an inverse. 
March 8th, 2017, 10:05 AM  #7 
Global Moderator Joined: Dec 2006 Posts: 18,034 Thanks: 1393 
3x² + x

March 8th, 2017, 05:31 PM  #8 
Newbie Joined: Sep 2016 From: Maharashtra Posts: 8 Thanks: 0 
So if I want to find its inverse in an equation form, what would be my approach? Could u please tell me.

March 8th, 2017, 10:34 PM  #9 
Senior Member Joined: Sep 2015 From: Southern California, USA Posts: 1,481 Thanks: 744  
March 8th, 2017, 11:37 PM  #10 
Global Moderator Joined: Dec 2006 Posts: 18,034 Thanks: 1393 
I just looked for a quadratic and found one. Once a solution is known, others follow as 3x(x  1)(x + 1) ≡ 0 (mod 9). 

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