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.
Best Answer
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".