Show $f:\mathbb{R}\rightarrow\mathbb{R}$ $x\rightarrow -x$ is bijective

Question

We have a function $$f:\mathbb{R}\rightarrow\mathbb{R}$$ $$x\rightarrow -x$$

Show $f$ is bijective.

One-to-one: Assume $f(x)=f(y)$ for some $x,y\in\mathbb{R}$, then $-x=-y$, hence $x=y$. Thus, $f$ is one-to-one function.

Onto:Let $y\in\mathbb{R}$. We need to find $x\in\mathbb{R}$ such that $f(x)=y$. Since $y\in\mathbb{R}$, let $-x=y\in\mathbb{R}$, so there is a $x\in\mathbb{R}$ such that $f(x)=-x$. Can you check?

Answer 1

One-to-one is fine.

For onto, take an arbitrary $y\in\Bbb R$. We see that $f(-y)=y$. This shows that $f$ is onto.

Answer 2

Alternatively:

Since $f$ is decreasing it is injective and since it is continuos it is also surjective.