The domain of a function is all the values of x for which f(x) yields real values.
The values of f(x) when x lies in the domain is the range.
Here x > 0 and f(x) = 1 - 3x
The range is all numbers less than 1.
We'll have to determine the set of real numbers that variable x can take, such as the expression that defines the function to be real.
We'll set f(x) = y = 1 - 3x
We'll move the term in x to the left and the term in y to the right:
3x = 1 - y
We'll divide by 3:
x = (1-y)/3
We know, from enunciation, that x>0, therefore (1-y)/3 > 0.
We'll solve the inequality:
(1-y)/3 > 0
1 - y > 0
-y > -1
We'll multiply by -1 and we'll change the direction of inequality:
y < 1
Therefore, the range of the function f is the opened interval (-`oo` ; 1).