Let $$y \propto \frac{1}{x}$$
Then we know y is inversely proportional to x. Then,
Why is y is inversely proportional to x same as $y \propto \frac{1}{x}$.
My attempt to prove this:-
Let $k$ be the constant of proportionality.
Then, if $y \propto x$ then multiplying k by x would increase y at the rate of k.
But, What if x is negative?
And also dividing k by x would reduce y.
But in the above statement I am not sure why would it reduce it by the same rate?
And also is it possible to x be a fraction and y increase?
I don't think so.
By definition, the inverse of $\space x\space$ is $\space\dfrac{1}{x}\space$ also shown as $\space x^{-1}.\space$
In humanspeak, that means saying
"$\space y\space $ is proportional to the inverse of $\space x$" $\quad$ is the same as saying
"$\space y\space $ is inversely proportional to $\space x.$"
For the staement about negative x, proportionality refers to magnitude, independent of sign.
In your statement about $k, \space \space k \space$ does not affect proportionality except for the first iteration. The ratio $y:xk\space$ always stays the same.