Have this function:
$$f(x) = \frac{x}{\sqrt{x^2+2x}}$$
And for whatever reason I decide to divide both the numerator and denominator by $x$:
$$f(x) = \frac{1}{\sqrt{1+\frac{2}{x}}}$$
My understanding is that the function is equivalent because dividing both parts by the exact same value doesn't change it algebraically.
And yet, when I look at the graphs, something strange happens with minus infinity:
The original:
The modified one:
The positive domain is the same it seems – it only appears flipped in the negative domain. My original understanding was that if the functions were algebraically the same, the graphs should be identical. But it appears that my division messed it up somehow.
Why is the negative domain flipped because of my division?
Best Answer
Because, for all $x\in(-\infty, -2)\cup (0,\infty)$, $\sqrt{x^2+2x}=\lvert x\rvert\sqrt{1+\frac2x}$, as opposed to your suggestion $x\sqrt{1+\frac2x}$.