For the record, I am sorry, I haven't yet learnt how to use LaTeX
I have a function $f(x) = 2x^3 – 1$
My proof of injection is as follows:
$f$ is one to one for all $x_1,x_2$ element $X$, if $f(x_1) = f(x_2)$ then $x_1 = x_2$
Proof
$f(x_1) = f(x_2)\\
2x_1^3 – 1 = 2x_2^3 – 1\\
2x_1^3 = 2x_2^3\\
x_1^3 = x_2^3$
Therefore $x_1 = x_2$
so $f(x)$ is one to one by direct proof – contraposition of 'if $x_1\neq x_2$, then $f(x_1)\neq f(x_2)$.
I am unsure how to approach the problem of surjection. I understand the concept, and I can show that it has a domain and a range which is an element of the real numbers, so it is definitely onto, but I don't know how to prove it.
Best Answer
You need to take $y\in\Bbb R,$ and show that there is some $x\in\Bbb R$ such that $f(x)=y$. In other words, you should show that $$y=2x^3-1$$ has a real solution $x$.