
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
Riley
OK.
Devyn
Is this function equal to $g(x) = x-1$?
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 $x=2$? In this case
Riley
Right, $f(2)$ is undefined because we cannot divide by zero. Hmm. Now I see the problem. Yikes!
In the context above, are $f$ and $g$ the same function?
yes no
Suppose $f$ and $g$ are functions but the domain of $f$ is different from the domain of $g$. Could it be that $f$ and $g$ are actually the same function?
yes no
Can the same function be represented by different formulas?
yes no
Are $f(x) = |x|$ and $g(x) = \sqrt {x^2}$ 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 $f(x) = \sin ^2(x)$ and $g(u) = \sin ^2(u)$. 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 $f = g$. The functions are equal. If $x \neq u$, then $f \neq g$. We have $f \neq g$ since $f$ uses the variable $x$ and $g$ uses the variable $u$.