Same or different?

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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
\[ f(x) = \frac {x^2 - 3x + 2}{x-2}. \]
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 \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: Sure! But what about when $x=2$? In this case
\[ g(2) = 1\qquad \text {but}\qquad f(2) \text { is undefined!} \]
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

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).

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 $u$.

2025-01-06 19:54:51