Let's take the function $$f(x)=\frac{x}{\sqrt{1-x^{2}}}.$$
My question is, why is the range of the function is all real numbers?
Because doesn't the fact that the denominator must be $f(x)=\sqrt{1-x^{2}}$ and that the numerator must be $x$ limit the amount of values the function can produce? Because for every $x$, only one denominator value is possible. Doesn't this limit the output of this function, therefore preventing it from producing all real numbers? And furthermore, can you prove to me that the $x$ inputs needed to produce every single real number are ordered from smallest to largest? Basically, why is it the case here that the larger the $x$-value you put in, the larger the $y$-value?
Can someone please explain to me, as simply as possible and without calculus, why the range of the function is all real numbers? And furthermore, can you prove to me that the $x$ inputs needed to produce every single real number are ordered from smallest to largest? Basically, why is it the case here that the larger the $x$-value you put in, the larger the $y$-value?
Best Answer
We have that $f(x)\ge 0$ for $x\ge 0$ and $f(x)< 0$ for $x< 0$ and thus
$$y=\frac{x}{\sqrt{1-x^2}}\implies y^2(1-x^2)=x^2\implies x^2(1+y^2)=y^2\implies x=\frac{y}{\sqrt{1+y^2}}$$
which is defined for any $y\in \mathbb{R}$.