What does "increases in proportion to" mean? I came across a multiple choice problem where a function $f(x) = \frac{x^2 - 1}{x+1} - x$ is given. One has to choose the statement that is correct about the function. The different statements about the function included:


*

*(1) the function increases in proportion to $x^2$.

*(2) the function increases in proportion to $x$

*(3) the function is constant.


Obviously the function is constant, but my questions is what the "increases in proportion to" means. I haven't come across it before, and was wondering if it is standard for something. I would like so see an example of a function that can be said to satisfy this statement.

To answer your question about examples where $f(x)$ would be proportional to $x$, and $x^2$, we only need to slightly modify the original function.
$f(x) = \frac {x^2-1}{x+1} $ is interesting between $x=-2$ and $x=2$, but as $x$ becomes really large or really small, the "-1" term and the "+1" term are insignificant and the expression is approximately $\frac {x^2}{x}$ or $x$. This would be proportional to $x$.
If $f(x) = \frac{x^3-1}{x+1} - x$, then as $x$ becomes really large or really small, the function breaks down to $\frac{x^3}{x} -x $ or $x^2$. This would be proportional to $x^2$ for much of $x$.
