Domain is usually a pretty straightforward concept, and is mostly just solving equations. However, one place I have found that people tend to make mistakes in domain is when they need to evaluate compositions.

For instance, consider the following problem:

#f(x) = sqrt(4x+1)#

#g(x) = 1/4x#

Evaluate #f(g(x))# and #g(f(x))# and state the domain of each composite function.

#f(g(x))#:

#sqrt(4(1/4x)+1)#

#sqrt(x+1)#

The domain of this is #x≥-1#, which you get by setting what's inside the root greater than or equal to zero.

#g(f(x))#:

#sqrt(4x+1)/4#

The domain of this is all reals.

Now if we had to combine the domains for the two functions, we would say that it is #x≥-1#. However, this is slightly wrong. This is because you need to consider the domain of each of your initial functions as well, which is something that people miss often. The domain of #1/4x# is simply all reals, but the domain of #sqrt(4x+1)# is #x≥-1/4# (which you get by setting everything under the radical# ≥ 0#).

Now, we know that the domain of everything put together is in fact #x≥-1/4#. This is one of the things I have seen other students miss pretty often.

Hope that helped :)