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There is a question in my calculus 101 textbook that says find the domain of the function. The function formulas are given, eg:

$$f(x) = \frac{x+4}{x^2 -9}$$ and $$g(t) = \sqrt[3]{2t - 1}$$

No other information or context is provided.

The back of the text says that the answers are $$(-\infty, -3)\cup(-3,3)\cup(3,\infty)$$ and $$(-\infty, \infty)$$

The answer for $g(t)$ makes some sense because, without any specification otherwise, it seems reasonable to me to assume that the domain is infinite. However, I don't understand the meaning of the answer to $f(x)$, nor do I understand how it can depend on the formula of the function. I was under the impression that domains are either defined explicitly and independently of the formula, or are undefined. I thought that the domain is the set of possible inputs to the function, and that therefore the domain is not at all dependent on (or constrained by) the formula.

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What textbook is this? – Dan Brumleve Sep 9 '11 at 7:51
@Dan Calculus: Early Trancendentals. 7th edition. Page 21. – Matt Munson Sep 9 '11 at 7:59
I think this is one of those "you just gotta drink the Kool-Aid" things... – Mehrdad Sep 9 '11 at 10:25
up vote 1 down vote accepted

What does a domain of a function mean in this context? The domain* represents the set of numbers where, in essences, the function "makes sense". In calculus you generally consider functions on the real numbers, which map elements of the set of Real Numbers ($\mathbb{R}$) to other elements of the set of real numbers. There are plenty of familiar functions that map other sets of numbers. For instance the factorial function $f(x) = x!$ maps integers to other integers, $f(1.5)$ does not make sense. If we were to consider the factorial function as a function on the real numbers, we would need some way to specify that only a subset (The integers) of the real numbers are actually legal arguments for the function.

For the function you are given, observe that the denominator $x^2 - 9$ is equal to 0 at $x = \pm3$. Division by zero is not defined in this context, and therefore $f(\pm3)$ as a function on the real numbers does not make sense. We need to specify what subset of the real numbers is the function valid, that is, all real numbers except for $\pm3$ - which is exactly what the answer indicates.

The Range of the function could also be limited. In the second example, $g(x) = \sqrt[3]{2t-1}$, as you correctly observed, All real numbers are valid arguments (hence the Domain is the entire set of Real numbers) and all real numbers can come out the other end. However, consider $h(x) = \sqrt{x^2}$, the domain for this function is still all the real numbers, however, $h(x)$ is always greater than 0. Hence while all the real numbers are valid input parameters, they can only be mapped to the positive real numbers. We use the set notation to indicate the fact that the output set is limited to a subset of $\mathbb{R}$

Challenge Question

What is the domain and range of $h(f(x))$? what about $f(h(x))$?

*To clarify, the domain is part of the definition of the function and is usually specified when the function is defined. In this case though, textbook authors have overloaded the term. The functions are assumed to be defined on the real numbers, and you are asked to determine which exact subset of the real numbers constitutes the domain.

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Rather, we are asked to determine the maximum subset of the real numbers which may constitute the domain. And to define what that means, we need Zorn's lemma. – Dan Brumleve Sep 9 '11 at 7:57
What's this about $g(x) = \sqrt(x)$? The function given was $g(t) = \sqrt[3]{2t - 1}$, with a cube root, which both takes negative arguments and yields negative values. – joriki Sep 9 '11 at 8:13
@jorki, you are right, its a little late here so I confused the example in my head with the actual example given in the OP. – crasic Sep 9 '11 at 8:16
@Dan Brumleve: In what sense could one need Zorn's Lemma to know the meaning of maximum subset? – André Nicolas Sep 9 '11 at 9:03
You might want to mention the difference between codomain and range. – Mehrdad Sep 9 '11 at 10:25

You're quite right; the book is using the term "domain" incorrectly. What they mean is "the greatest possible subset of the real numbers that could be used as the domain of a function whose values are given by this formula".

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As you say, the domain of a function is part of its definition. The "definitions" of $f$ and $g$, in the context of the question, are not definitions at all, but merely equations. I would reformulate the question as: what is the largest domain in $\mathbb{R}$ for which this equation defines a function?

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