Finding the derivative of rational function with unknown constant in numerator and denominator

Question

I'm asked here to find the domain and differentiation range for the following function: f(x)= (x-a)/(x+a) with a>0 When plugging it into a graphing calculator, I see that it's a similar function to 1/x but I have no idea how to get there from the given formula. Is there factoring involved?

Answer 1

You can use $f(x) = 1- \frac{2a}{x+a}$, from which you get $f'(x) = \frac{2a}{(x+a)^2}$. The domain of this function is clearly $\mathbb{R} \backslash \{-a\}$, and it ranges over all numbers that have the same sign as $a$.