Prove that the following function is one to one

Question

I want to show that the function $$f(x)=\frac{3-8x^{^{2}}}{1-x^{3}}$$ is one to one. For this, I suppose that $f(x)=f(y)$. So $$(x-y)(3(x^{2}+xy+y^{2})-8(x+y)-8x^{2}y^{2})=0$$ If show that $$3(x^{2}+xy+y^{2})-8(x+y)-8x^{2}y^{2})\neq0$$, then we deduce $x=y$. Please help me.

Answer 1

It isn't one to one. For example $f\left(\dfrac{\sqrt3}{\sqrt8}\right)=f\left(-\dfrac{\sqrt3}{\sqrt8}\right).$

Is one to one (with the derivative test) if its domain is restricted to $(-\infty,-1.553\ldots)$ or $(-1.553\ldots,0)$ or ...

Answer 2

You can use a function grapher to plot this function. It is obvious from the graph that this is not a one-to-one function, if you do not put any restriction on the domain.