Determine if function $f(x)=\frac{1-2x}{x-2}$ is invertible. If it is, show the inverse function.
I need help with proving surjectivity of the function.
The domain is $\mathbb{R} \setminus \{2\}$, since $x-2 \neq 0 \implies x \neq 2$.
Range is $\mathbb{R} \setminus \{-2\}$, because of:
$ y=\frac{1-2x}{x-2}\\ yx-2y=1-2x\\ yx+2x=1+2y\\ x(y+2)=1+2y\\ x=\frac{1+2y}{y+2}\implies y \neq -2 $
$f:\mathbb{R} \setminus \{2\} \rightarrow \mathbb{R} \setminus \{-2\}$
How can I prove a surjection?
Perhaps you mean injectivity because you have shown surjectivity?
Since if you fix any $y \in \mathbb{R} \setminus \left\{2\right\}$, then for $x \ne 2$: $$y=\frac{1-2x}{x-2}\iff x=\frac{1+2y}{y+2}$$
The function is also injective because $f(a)=f(b)$ implies $a=b$; since for $a \ne 2$ and $b \ne 2$: $$\begin{align}\frac{1-2a}{a-2}=\frac{1-2b}{b-2} & \iff (1-2a)(b-2)=(1-2b)(a-2) \\ & \iff b-2-2ab+4a = a-2-2ab+4b \\ & \iff b+4a=a+4b\\ & \iff a=b\end{align}$$