Let f(x)=(x-9)/(3x+1). Let g(x)=(5x+6)/(2x+11). Find (f/g)(x) and its domain. Is it any way I can learn this, I am really trying...
The domain of any rational function is all reals, except where the denominator is zero.
so, the domain of f is all reals except where 3x+1 = 0.
The domain of g is all reals except where 2x+11 = 0.
f/g (x-9)/(3x+1) / ((5x+6)/(2x+11))
= (x-9)/(3x+1) * (2x+11)/(5x+6)
Now, you might think that the domain is all reals except where 3x+1 = 0 or where 5x+6 = 0. These values are indeed excluded, but you also have to exclude 2x+11 = 0, since f/g is defined in terms of f and g, and g is not defined where 2x+11 = 0!