# Function$f(x)={2x+3xf(x)\over x-1}$ is one to one and an onto function. what is the codomain of f(x)?

$$f(x)={2x+3xf(x)\over x-1}$$ This function is one to one and an onto function. what is the codomain of $$f(x)$$? How to solve this function?

$$y(x-1)=2x+3xy\implies y = {-2x\over 2x+1}$$
so it is linear rational function so the range is $$\mathbb{R}-\{-1\}$$
The range of a linear rational function is $$f(x)= {ax+b\over cx+d}$$ is $$\mathbb{R}-\{{a\over c} \}$$.
Proof: Let $$y_0$$ be in a range of $$f$$, so there is an $$x_0$$ such that $$y_0= {ax_0+b\over cx_0+d}$$ then $$y_0cx_0+y_0d = ax_0+b$$ so $$x_0(cy_0-a)=b-y_0d\implies x_0 = {b-y_0d\over cy_0-a} \;\;\;{\rm if}\;\;y_o\ne {a\over c}$$ and thus a conclusion.