What is the domanin and inverse function of (ax^2+bx+c)/(dx^2+ex+f) ???

Question

What is the DOMAIN & INVERSE FUNCTION of $\frac{a x^{2} + b x + c}{{d x}^{2} + e x + f}$ $(ax^2+bx+c)/(dx^2+ex+f)$ ?

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Answer 1 · 2017-06-09T22:50:30

See explanation...

Given:

$f (x) = \frac{a x^{2} + b x + c}{{d x}^{2} + e x + f}$ $f(x) = (ax^2+bx+c)/(dx^2+ex+f)$

I will assume that $a \neq 0$ $a != 0$ and $d \neq 0$ $d!=0$

Note that this function is well defined except where the denominator is $0$ $0$ . We can use the quadratic formula to express this as:

If $e^{2} - 4 d f < 0$ $e^2 - 4df < 0$ then the domain of $f (x)$ $f(x)$ is $RR$
If $e^2 -4df >= 0$ then the domain of $f(x)$ is $RR "\" { (-e+-sqrt(e^2-4df))/(2d) }$

In either case, the inverse relation of $f(x)$ is not a function.

$f(x)$ is asymptotic to $y = a/d$ for large positive and negative values of $x$ and the inverse relation fails the vertical line test.