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4

At A-level (16+ in the UK) students are never asked questions such as "what is the domain of $f(x)=x^2$ in formal examination questions. In my teaching I mention that such a question is meaningless, the domain is part of the definition of the function and the function isn't fully defined until a domain is given. Question papers are careful always to ...

3

At the secondary level, students tend to think of "real numbers" as being literal: any number that isn't in the real number system isn't a "real number", and any time you're given a variable, the default is that it can be any real number, and the default is that a function applies to all real numbers. The idea of a function that takes ...

9

In my high school days (1970s), the set of all numbers for which an expression is defined (and, implicitly, real) was called the “natural domain” of the function defined by the expression. The domain of a function with an unspecified domain was to be understood to be its natural domain, a convenient assumption. I’m not sure this is an accurate answer to why ...

13

We give high-schoolers many different explanations of the word "function." Here are a few that are either implied or outright stated at various points in a student's education: A function is an expression in terms of $x$. This is pretty unusual all by itself, but it may appear in conjunction with (2) or (3) (i.e. students may reason that $f(x)$ or ...

21

In real-world applications, the typical case is that the domain is neither implicit in an expression we write down, nor explicitly stated along with the expression. Rather, one uses knowledge of the real world to decide what numbers make sense as inputs to the function. So for example, if $x$ represents an investment, and $f(x)=x-3$ represents the profit on ...

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