g(t) = sin (e to the power of -t)?

Question

Find the domain of the each function

Answer 1

The domain is the set of input values the function can have. IN cases where you are given a function whose domain is not explicitly specified, you find the largest subset of the universe of discourse (that is, the largest set of numbers that your class deals with when considering the domain and range of functions -- this is probably the real numbers, although it might be the complex numbers, depending on the class) for which the function definition actually makes sense. In this case, the function sin (e^(-t)) is defined for all real numbers, so you would report the domain as R (or if the universe of discourse is C, then the domain is C, since this function is in fact defined for all complex numbers by its power series).

Dr. D, please review the definitions of the words "domain" and "range." They are not the same thing.

Answer 2

At a glance, e^(-t) has the domain of R, with the range of {x in R: x > 0}.
Sin(e^-t) has the domain equal to the range space of e^-t, or
{x in R: x>0}.

Answer 3

Write the function as

g(t) = sin [e^(-t)] or

g(t) = sin [1 / (e^t)].

e^t is greater than zero for all real numbers.

Since e^t is never zero, 1/ (e^t) is defined for all real numbers and
sin[1 / (e^t)] is defined for all real numbers.

Answer: The domain of g(t) = sin [1 / (e^t)] is all real numbers.

Answer 4

Assuming t takes real values
then e^(-t) ranges from 0 to infinity - always positive.
sin of anything ranges from -1 to +1

So g(t) ranges from -1 to +1