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 $f(x) = \frac {2x}{x}.$
Riley: OK.
Devyn: Is this function equal to $g(x) = 2$ ?
Riley: Well if I plot them with my calculator, they look the same.
Devyn: I know!
Riley: And I suppose if I write $\begin{align*} f(x) &= \frac{2x}{x} \\ &= 2 \cdot \frac{x}{x} \\ &= 2 \cdot 1 \\ &= 2 \\ &= g(x). \end{align*}$
Devyn: Sure! But what about when $x=0$ ? In this case $g(0) = 2\qquad \text {but}\qquad f(0) \text { is undefined!}$
Riley: Right, $f(0)$ 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 $f(x) = 2x+1$ and $g(x) = 2(x-1)+3$ 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 $f(x) =x^2$ and $g(u) = u^2$ . 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 $u$ .

Although one function is stated in terms of $x$ and the other is stated in terms of $u$ , they are both fundamentally the same function: $f(1) = g(1)$ , $f(-4) = g(-4)$ , and, in general $f(c) = g(c)$ for every value of $c$ in the real numbers. You could also graph both of these functions, and they would have the same shape. The mathematical relation $f(x)$ conveys is the same one that $g(u)$ conveys, so we conclude that $f = g$ .