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

Question

What is the DOMAIN & INVERSE FUNCTION of #(ax^2+bx+c)/(dx^2+ex+f)# ?

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Answer 1 · 2017-06-09T22:50:30

See explanation...

Given:

#f(x) = (ax^2+bx+c)/(dx^2+ex+f)#

I will assume that #a != 0# and #d!=0#

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

If #e^2 - 4df < 0# then the domain of #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.