I'm a software architect, working on some algorithms involving unique storage formats for large numbers.
The function I'm working with is:$$y = \frac{mx^p+b}{d}$$ ... where $p$ and $d$ would traditionally $=1$ for standard linear graphs.
However, for the algorithms I'm working on, $p$ may be any integer (including $0$ and negatives), and $d$ may be any integer $\ge0$ (with $0$ resulting in $y$ being undefined, obviously).
$x$ may be any rational number $\ge0$
$m$ and $b$ may be any integer
In this context, are there any consistent, universal shortcuts for predicting whether $y$ will be negative or positive (or undefined for other reasons)?
I've been researching x-intercepts, but I'm not sure where I need to go with this.
I think it boils down to whether $mx^p\gt-b$, but I need to make sure there aren't any nuances with the way $m$, $x$, $p$, and $b$ interact on the whole.
If I can predict certain details of $y$ based on properties (or ranges) of $x$ (and/or $p$ etc.), without having to actually compute the entire function, that would be a huge benefit for optimization purposes, given the algorithms we are using.
I really need to avoid computing the exponent in particular.
$d$ has no effect on the sign of $y$, as long as it is not equal to $0$ (and if it is, you have bigger problems anyway).
The value of $y$ will be positive if and only if $mx^p + b>0$ which is true if and only if $$mx^p>-b,$$ as you correctly observed.
Now, if $m>0$, this means that $y$ is positive if and only if $$x^p>-\frac bm.$$
If $b$ is also positive, then this is always true (since $x\geq 0$), and if $b$ is negative, then $-b=|b|$, and then $y$ is positive if $x>\sqrt[p]{\frac{|b|}{m}}$
If $m<0$, then the $>$ sign flips when you divide by $m$ and you get that $y$ is positive iff $$x^p<-\frac bm$$
This has no solution if $b<0$ and has a solution similar to the one above if $b>0$.