I need an alternate proof for this problem.
Show that the function is one-one, provide a proof.
$f:x \rightarrow x^3 + x : x \in \mathbb{R}$
I needed to show that the function is a one-one function. I tried doing $f(x) = f(y) \Rightarrow x = y$, It ended up with $x(x^2 + 1) = y(y^2 + 1)$. I couldn't figure out how to solve this further.
Instead I had good idea of what the graph looked like, so I tried to show that it is a strictly increasing function.
$f(x_2) - f(x_1) = (x_2^3 - x_1^3) + (x_2 - x_1)$
Thus, $f(x_2) - f(x_1) > 0$
Hence, f(x) is strictly increasing. And hence is one-one.
Unfortunately this solution isn't acceptable. :( Can you guys help me work out how to do it the right way. Thanks again for your help.