TITLE: What makes this answer unsimplified?
QUESTION [9 upvotes]: Generally, what are the rules for simplifying answers?
I had this question:
Given $f(x) = \frac{1}{x}$, evaluate $\frac{f(x) - f(a)}{x-a}$.
Why is the following a bad answer:
$$\frac{\frac{1}{x} - \frac{1}{a}}{x-a}$$
The better answer is:
$$\frac{-1}{xa}$$
Why is this? Is it generally just bad to leave fractions in either the numerator or denominator? So we should try to eliminate the fraction in the numerator by solving the equation first on the top by using a common denominator?

REPLY [20 votes]: I like your bad answer better because $$\frac{\frac{1}{x}-\frac{1}{a}}{x-a}=-\frac{1}{ax}$$ is true only if $x\ne a.$ 
For $x=a$ the left hand side is undefined while the right hand side is defined unless $a=0$