Is (0,0) a Point on the Graph of y=x^-1?

[Mentallic]

If I'm asked to graph this function: $$y=\frac{1}{x^{-1}}$$

Is x=0 undefined? Obviously by the rule of powers, this equation is the same as y=x, but I'm unsure if the point (0,0) exists in this equation or not.

 

[Tinyboss]

That function is not defined at x=0. To simplify it to x, you rely on the fact that you can multiply by 1=x/x. But x/x isn't defined when x=0, so you can't use simplification to get around the undefinedness at 0.

 

[Mentallic]

Yes, if I converted the power to a fraction as so: $$\frac{1}{\frac{1}{x}}$$ then I'd be relying on that rule, but what about if I used the rule of powers, i.e. $$\frac{1}{x^a}=x^{-a}$$ So simply, $$\frac{1}{x^{-1}}=x^{-(-1)}=x$$

It just seems to me that only sometimes this is undefined, depending on how you treat the problem.

Sort of like $$\sqrt{x^2}=|x|$$ while $$(\sqrt{x})^2=x$$ and defined for only $x\geq 0$

 

[Tinyboss]

That rule explicitly requires $$x\ne0$$.

 

[Mentallic]

Ahh yes, of course!

Thanks tinyboss