Hey so, I jumped a math class and I have a big test coming. I know how to find the domain of a function without a graph, but how do you find the range? The range of a function without a graph, I mean. They discussed it in the class before this one but I wasn't there and the notes aren't telling me much. If there isn't a specific way, are there any tips or tricks? Specifically quadratic functions or functions with square roots in them? My teacher in his notes just replaces the radicand or any other number in a parentheses like thing with [0, infinity)

1. f(x) = x-2, x ≤ 3

Then,

f(x) ≤ f(3)

f(x) ≤ 3-2

f(x) ≤ 1

It leads to the correct answer.

2. g(x) = 2x², -3 ≤ x ≤ 1

Then,

g(-3) ≤ g(x) ≤ g(1)

2(-3)² ≤ g(x) ≤ 2(1)²

18 ≤ g(x) ≤ 2

In this case, of course, it makes little sense (at least to me). Of course, by drawing a graph, you easily get the correct answer, 0 ≤ g(x) ≤ 18.

I have two questions:

1. Why does this not work? What's really going on?

2. Is it possible to do without drawing a graph? How?