Two young mathematicians examine one (or two!) functions.

Check out this dialogue between two calculus students (based on a true story):
Devyn
Riley, I have a pressing question.
Riley
Tell me. Tell me everything.
Devyn
Think about the function
Riley
OK.
Devyn
Is this function equal to ?
Riley
Well if I plot them with my calculator, they look the same.
Devyn
I know!
Riley
And I suppose if I write
Devyn
Sure! But what about when ? In this case
Riley
Right, is undefined because we cannot divide by zero. Hmm. Now I see the problem. Yikes!
In the context above, are and the same function?
yes no
Suppose and are functions but the domain of is different from the domain of . Could it be that and are actually the same function?
yes no
Can the same function be represented by different formulas?
yes no
Are and the same function?
These are the same function although they are represented by different formulas. These are different functions because they have different formulas.
Let and . The domain of each of these functions is all real numbers. Which of the following statements are true?
There is not enough information to determine if . The functions are equal. If , then . We have since uses the variable and uses the variable .