In general:
For any
function f(x), the range is all the numbers that
f(x) can possibly
output.
Let's say that
f(x) = 3x-15. Now, for any
real number you can think of, there's an x value than we can plug into
f(x) to get that number. So we say that the range of
f(x) is all real numbers, or, in
set notation, {
f(x):
f(x)
∈R}.
For our second example,
g(x) = 3/(x-1). Now, since the
numerator of 3/(x-1) is positive, then there's no value for x-1, and therefore no value for x, that will make the function equal 0. So the range of
g(x) is all real numbers save for 0: {
g(x):
g(x)!=0}. (In this case,
g(x) being an element of the real numbers is implied.)
One more:
h(x) = sqrt(x). Now, remember that putting any negative x into this function will return a
nonreal answer, which would be a
Bad Thing. Subsequently, there is no real x which, when put into
h(x), will output a negative answer. The range of
h(x), therefore, is all numbers greater than or equal to 0, or {
h(x):
h(x)>=0}.
Contrast
domain.