Rational function analysis

— 1 minute read

Set-up

We consider the rational function f:x↦ax2+bx+cdx+e,x∈R,x≠−ed f: x \mapsto \frac{ax^2 + bx + c}{dx + e}, \qquad x \in \mathbb{R}, x \neq - \frac{e}{d}

where a,d≠0a,d \neq 0.

Under what conditions is the range of ff, Rf=RR_f = \mathbb{R}? If Rf≠RR_f \neq \mathbb{R}, what is its range?

Results

This question depends on the sign of the special quantity we will call the special discriminant and denote by ss.

s=a(ae2−bde+cd2) s = a(ae^2 - bde + cd^2)

If s=0s = 0, then the function 'collapses' to a linear function.

If s<0s < 0, then the function has range Rf=RR_f = \mathbb{R}.

If s>0s > 0, then Rf=(−∞,bd−2ae−2sd2]⋃[bd−2ae−2sd2,∞)R_f = \left( -\infty, \frac{bd-2ae-2\sqrt{s}}{d^2} \right] \bigcup \left[ \frac{bd-2ae-2\sqrt{s}}{d^2}, \infty \right)