The two terms come from discrete mathematics, indeed from maps. If x maps to y there is an association for values of x to values of y. One would write m as a map :-

m = [x->y]

Ie, a set of mappings. Then the domain of the map is the set of x values and the range the set of y values :-

Dom(m) = [x], Range(m) = [y]

So applied to "continuous functions" the domain of the function is the values for which the function exists, the range the values which the function can take.

For f(x) = x, the domian and range are the same, [-oo,+oo]

In f(x) = x² the domain is still [-oo,+oo] (since for all these possible values the function has a result), but the range is [0,+oo] since no negative number may be a square.

In f(x)=sqrt(1-x²) the domain and range are [-1,+1], ie: confined within the unit circle centered on the origin.

So, the best way to calculate domain and range is to draw a graph. This will show (y axis) which range values may be taken for which domian values (x axis).

m = [x->y]

Ie, a set of mappings. Then the domain of the map is the set of x values and the range the set of y values :-

Dom(m) = [x], Range(m) = [y]

So applied to "continuous functions" the domain of the function is the values for which the function exists, the range the values which the function can take.

For f(x) = x, the domian and range are the same, [-oo,+oo]

In f(x) = x² the domain is still [-oo,+oo] (since for all these possible values the function has a result), but the range is [0,+oo] since no negative number may be a square.

In f(x)=sqrt(1-x²) the domain and range are [-1,+1], ie: confined within the unit circle centered on the origin.

So, the best way to calculate domain and range is to draw a graph. This will show (y axis) which range values may be taken for which domian values (x axis).