
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 $$.