Question

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

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

Answer 1

See explanation...

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.