CONDITIONALS: IF/WHETHER

In many cases, if and whether can be used interchangeably without affecting the meaning.  

IF
In the following situations, speakers prefer using IF.
REPORTED SPEECH
(I asked them, "Are you leaving?") I asked them if they were leaving.
SLIGHTLY LESS FORMAL
I asked him if he is going to visit. (informal context) (I asked him whether he would visit. (more formal))
CONDITIONAL
I asked him questions if he was not too busy. I asked him questions when he was not too busy.

WHETHER
In the following situations, speakers prefer using whether.
WITH WORDS USED IN FORMAL CONTEXTS
We inquired whether the President would attend the summit meeting.  (investigate, explain, examine, study, decide, determine)
WHEN STRESSED IN SPEECH
We must ask ourselves whether we will accept failure.
BEFORE "OR NOT" / REGARDLESS
We are leaving whether you like it or not.  (no option exists)
DOUBT + "OR NOT" / ALTERNATIVE
We don't know whether we are going or not.  (doubt, can't say, be uncertain— about an alternative)
BEFORE AN INFINITIVE
We can't decide whether to leave or to stay.
AFTER A PREPOSITION
We were worried about whether you would like it.
AFTER "THE QUESTION IS…"
The main question is whether they are going also. (the problem is, the undecided point is)
INITIAL POSITION / SUBJECT
Whether my cell phone works there is a mystery to me. (no comma required, as the clause functions as a subject).