<s>[INST] how to show cubic function is one to one using assumption Okay I know the function $f(x)= x^3 +2x+3$ is one to one but how do I show that using the assumption that $f(a)=f(b)$ where $a=b$? 
So if I do the process I have 
$ a^3+2a+3 = b^3+2b+3$ my 3's cancel and i get
$a^3+2a= b^3+2b$
I know I can factor out an $a$ and $b$
$a(a^2+2)=b(b^2+2)$
but what do I do from here to show $a=b$? 
 [/INST] Notice that
$$a^3+2a= b^3+2b\iff a^3-b^3+2(a-b)=0\iff (a-b)(a^2+ab+b^2+2)=0...(1)$$
Since $$a^2+ab+b^2+2=\left(a+\frac{b}{2}\right)^2+\frac{b^2}{4}+2\ge 2$$
it follows that the only real solution of ($1$) is $a=b$. Then $f$ is one-to-one.

Another approach:
Since $f'(x)=3x^2+2>0$ for any $x\in\mathbb{R}$ it follows $f$ is increasing, then monotone, which implies it is one-to-one.
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