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Question 322744: I know the differences between domain and range, and that square roots are a restricted function, but are there others?
Answer by solver91311(24713)   (Show Source):
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All domains are restricted in the sense that any value that would make the function undefined must be excluded. Some functions, most notably polynomial functions and exponential functions have no restrictions on the domain, which is to say that the domain is all real numbers.
You gave one example of a type of function which does have a restricted domain, namely a function that involves a radical with an even-numbered index, or equivalently, a fractional exponent with an even-numbered denominator. In this case, the domain must be restricted such that the radicand is always greater than or equal to zero.
Another common example is the rational function, such as \ =\ \frac{g(x)}{h(x)}) . In such a function the domain must be restricted such that \ \neq\ 0) .
The domain of Logarithmic functions must be restricted such that the argument of the log function is greater than 0. Note that the argument must be strictly greater than 0; it cannot equal zero.
Trigonometric functions are often artificially restricted because of their periodic nature, and the fact that some trig fuctions have discontinuities at 0,  or  .
And finally, where you have a complex function that includes more than one of these considerations, all must be taken into account.
John
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