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**deathlight** Aforl that is what I'm looking for but I still don't get it

Okay... Say, there's a function f(x) = 3x. 'Domain' refers to the set of permissible inputs. In this case, the input is x. So the domain of f(x) is therefore the set of x that can be entered into this function. Obviously the domain has no restrictions because x can be any real value and there is a corresponding f(x) for all x. Say, x = 4, f(x) = 12; x = -0.5, f(x) = -1.5 etc. Therefore, the range is all real values of f(x).

But of course you can choose to limit the domain to a particular set only. Say now, the domain of f(x) is restricted to 1 < x < 5. So now the range is therefore only 3 < f(x) < 15. The graph below shows the domain and the range of f(x), given by the grey and yellow portions respectively.

Now let's consider f(x) = x^2 - 1.

The domain can be x ∈ R (as in, x can be any real values). However in this case, it is obvious that f(x) cannot be less than 1. (Since x^2 must be positive). Therefore, the range is f(x) >= -1.

If you were to restrict the domain from -2 < x < 3, the corresponding range will therefore be -1 ≤ f(x) < 9. (see graph, when x varies from -2 to 3, the value of f(x) varies from -1 to 9)