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

Question

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

Answer 1

$$y(x-1)=2x+3xy\implies y = {-2x\over 2x+1}$$

so it is linear rational function so the range is $\mathbb{R}-\{-1\}$

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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.