I have a series of inequalities:
$$y_{1min} \leq y_{1}(x) \leq y_{1max}$$ $$y_{2min} \leq y_{2}(x) \leq y_{2max}$$ $$..$$ $$y_{nmin} \leq y_{n}(x) \leq y_{nmax}$$
Note that $x\in\mathbb{R}$
The question is, is there a general method that allows me to find the range of $x$ that satisfies the above inequalities?
Things are easy if there is just one inequality, and $y_1(x)$ is at most a quadratic function. But I am looking for a general solution here.
Edit: My $y$ function can roughly be written as
$$y=\Biggl[\frac{(x^2+bx+c)^f}{(dx+e)^g}\Biggr]$$
where $c$ and $e$ are small comparatively.$(dx+e)^g$ must be positive.
In one of the $y_i(x)$, $f=5/3, g=2/3$, but in another, $f=1, g=1$. It is safe to assume that both $f,g\in\mathbb{Q}$, that is they are rational numbers.
$b,c,d,e\in\mathbb{R}$ , and they vary from one $y_i(x)$ to another.
Hope this helps.
Edit:Let's make this question a little general; instead of letting $y(x)$ depends on one variable, what if $y$ depending on multiple variables? That is, how to find the range for $(x_1,x_2,x_3,..., x_i)$, given that
$$y_{1min} \leq y_{1}(x_1,x_2,x_3,..., x_i) \leq y_{1max}$$ $$y_{2min} \leq y_{2}(x_1,x_2,x_3,..., x_i) \leq y_{2max}$$ $$..$$ $$y_{nmin} \leq y_{n}(x_1,x_2,x_3,..., x_i) \leq y_{nmax}$$