select Function 4 and use the Range and Domain sliders to identify the range and domain. Complete parts 1 and 2 below. Use the interactive figure to find your answer. Use the left and right arrow keys to move along a slider as needed. Click here to launch the interactive figure.

The domain of a piecewise function is the union of the domains of each piece of the function. Similarly the range for a piecewise function is the union of the ranges for each piece of the function

Looking at the graph, we can see two subdomains. When x ≤ 2 , we have one subdomain and when x > 3 there is another subdomain. We state this as
D = - ∞ < x ≤ 2 or 3 < x ≤ 6
In interval notation we represent this as

D = (-∞, 2] U (3, 6]

The nature of the brackets is important a [ or ] bracket means that the value to which it is attached is included whereas an open bracket, ( or ) means that value is not included

For example, (-∞, 2] means that everything below x = 2 to -∞ is part of the domain but it does not include -∞; since at -∞, a function is not defined. However it does include 2 in the domain as can be seen by the closed bracket on the right

The other subdomain is for x > 3 (note: a filled circle means the value is included, an empty circle denotes the value is excluded)

The second subdomain is
3 < x ≤ 6
In interval notation, it is (3, 6]

So the total domain is the union of the two intervals

The range is also discontinuous and there are two distinct ranges. The range is nothing but all the set of values generated by the function. So we look at all possible values for y that the function produces

The Range of this function is -∞ < f(x) ≤ 1 or -1 < f(x) ≤ -4
In interval notation this is
R = (-∞ , 1] U (-1, 4]