Same or different? Take 2

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Two young mathematicians examine one (or two?) functions once again.

Check out this dialogue between two calculus students (based on a true story):

Devyn: Riley, I have another pressing question.
Riley: Sure, what is it?
Devyn: Think about the function $f(x) = \frac {x^2-3x+2}{x-2}.$
Riley: Okay. What about it?
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! But remember when we realized before that $\frac {2x}{x}$ and $2$ are not the same function? I think this could be a similar situation.
Riley: Okay, let’s check. I suppose I could write $\begin{align*} f(x) &= \frac{x^2-3x+2}{x-2} \\ &= \frac{(x-1)(x-2)}{x-2} \\ &= x-1 \\ &= g(x). \end{align*}$
Devyn: Right! But what about when $x=1$ ? In this case, $g(1) = 0\qquad \text {but}\qquad f(0) \text { is undefined!}$
Riley: Okay, $f(0)$ is undefined because we cannot divide by zero. Hmm...

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

The domain of a function is part of the ‘‘data’’ of the function. A function is not a rule for transforming the input to the output, but rather the relationship between a specified collection of inputs (the domain) and possible outputs (the range).

If you simplify a function $f$ and realize the simplified form is $g$ , is it possible that $f$ and $g$ have different domains and are therefore different functions?

yes no

When simplifying functions, you have to be careful because sometimes the simplified form of a function does not have the same domain as the original function. In this case, the two functions are not actually equal! We will investigate this further in the next card.