Does Inductive Reasoning Prove f(x) Equals x Squared for All Natural Numbers?

[Andrax]

This is from a functional equation , i think there is no points in posting it here this is what i ant to prove the functions is from N to N .

let x be a natural number
i want to prove that f(x)=$$x^2$$

By induction
suppose that f(x)=$$x^2$$
f(0)=0 holds (from the functional equation)

i want to prove that f(x)= $$(x+1)^2$$

when putting (x,1)(we have f(x+y)+f(x-y) in the equation) I need to know the value of f(x-1) ,does the assumption that f(x)=$$x^2$$, imply that f(x-1)=$$(x-1)^2$$
(this is not a homework i just saw this exercise online and decided to do it).

 

[mfb]

Where is the functional equation?
Do you have a question, or do you want to start a discussion (about what?)?

If you want to prove f(x)=x², you cannot just assume that this is true (without additional statement, this means "for all (natural) x"). It ruins the idea to prove it.

 

[Andrax]

i solved the issue now , thanks ,everything is clear,
( i meant for an unspecific x let f(x))=x^2 then i'd have to prove by induction ..