Next step to take in direct proof for one to one?

Question

This is from Discrete Mathematics and its Applications

And the definition of strictly increasing.

Here is my work so far.

I know that a direct proof involves making an assumption p, which in this case is that the function is strictly increasing, and then taking logical steps to reaching q, which is what I am trying to prove, in this case being -function is one to one.

I wrote down the assumption and am trying to take steps to get from that to the one to one definition. From intuition, I understand that if it's strictly increasing, no function outputs can be the same so there are no conditions broken for one to one. But what step should i take next in the direct proof to express this?

Answer 1

You are very nearly there. You just need to show that this function satisfies that definition.

So suppose that $f(x) = f(y)$ for some $x, y \in \mathbb{R}$. Then can we have $x \neq y$? Why/why not? Use the property that $x \neq y \implies f(x) \neq f(y)$ for this function!