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In my maths books it's said that: "when the domain of f(x) is not explicitly defined then in this case domain will mean the set of values of x for which f(x) assumes real value"

SO what it meant by domain to be implicitly/explicitly define

And what it the logic behind the above extratc...why we take real values only can't there be any other fields like C ?

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1 Answer

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Explicit means stated in an outward and purposeful manner. That is, they told you the domain. Implicit means defined in a tacit manner, where you understand what it is out of context of the parameters of the general scenario you're working in.

They are just making a convention here that when the domain is not stated, they mean the largest possible domain so that it's range is a subset of the reals (They'll state otherwise when it's not).

It's just a convention. The usual reasons behind conventions are desire for consistency and elimination of ambiguity which might occur when multiple people are communicating about the topic at hand.

Establishing a convention is useful because it gets everyone doing things the same way. It keeps things from getting confusing. Sort of like how there are two possible ways to define the cross product, and they just selected the one that abides by the right hand rule as convention so everyone will use the same definition, even though they could have just as easily defined it to abide by the left hand rule instead.

That doesn't mean that sometimes a convention isn't established for a reason: sort of like how we choose to use a basis to describe a vector space rather than just any old spanning set. That's because bases have additional (really nice) properties we're also interested in. We could equally well identify a vector space by any of its spanning sets. It just so happens that a basis has some really nice properties by virtue of it's linear independence, not to mention that it contains the minimum number of elements of the vector space we need in order to identify the vector space explicitly by defining it as the span of the indicated basis.

I hope that helps.

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