Domain and range of a functio: One of the more significant ideas regarding functions is that of the domain and range of a function. In simplest world the domain of function is the set of all values which can be plugged in a function and have the function present and have a real number for a value. Thus, for the domain we have to avoid division by zero, square roots of -ve numbers, logarithms of zero & logarithms of negative numbers, etc. The range of a function is just the set of all possible values which a function can take.
Let's determine the domain and range of a few functions.
Example : Determine the domain and range of following functions.
f ( x ) = 5x - 3
Solution
We know that it is a line & that it's not a horizontal line (Since the slope is 5 & not zero...). It means that this function can take on any value and thus the range is all real numbers. By using "mathematical" notation it is,
Range : ( -∞, ∞ )
It is more usually a polynomial and we know that we can plug any value in a polynomial and thus the domain in this case is also all real numbers or,
Domain: - ∞ < x < ∞ or (-∞, ∞)
On the whole determining the range of a function can be rather difficult. As long as we limit ourselves down to "simple" functions, some of which we looked at in the earlier example, determining the range is not too bad, however for most of the functions it can be a difficult procedure.
Due to the difficulty in determining the range for a lot of functions we had to keep those in the earlier set somewhat simple that also meant that we couldn't actually look at some of the more complicated domain instance that are liable to be significant in a Calculus course. Thus, let's take a look at another set of functions only this time we'll just look for the domain.