Domain and Range Question. Please help

Question

One end solid dot is (0,-1) and the other solid dot is (-2,1). One hollow dot is (-5,2) with the other hollow dot being (-2,4). Write the Domain and Range of f as intervals or unions of intervals.

 

Your help is appreciated

Answer 1

A hollow dot represents a part of the part of a function where f(x) is not defined. Simpler put, it is a number, x, plugged into the function such that the equation does not work. For example, in the function f(x) = x+1/x-1, the function works for all numbers, with the exception of 1. This is because if x is 1, the denominator (x-1) will be 0, and we cannot divide by 0. The function does not work at 1.

 

Applied to your problem, it is easiest if we plot the function out on a graph. If we do this, we see that the first dot is at (-5,2), and is hollow. This means the function is not defined at this point. The next point is at (-2,4) and is also hollow. For the next length of line, the first point is at (-2,1), and the second is at (0,-1). Both are solid, which means that they are defined at both points.

 

Now, we look at the domain first. The domain is the set of x's for which f(x) is defined, basically, all the points that exist going horizontally. In interval notation, ( is used next to a number that is not included in the domain. It doesn't matter how close a number gets to this noninclusive number, as long as it doesn't reach it, it's included in the domain. ] is used to denote a number that is included in the domain. The domain of this function will be (-5,0], meaning the domain runs from -5 to 0 including 0 and excluding -5. Note that -5 is excluded from the domain, because it has a hollow dot on the graph, and 0 is included, because it's dot is solid. It is also of note that -2, although it has a hollow dot in one location, is included in the graph, because there is another, solid, dot at -2 which joins the 2 sections. 

 

Range is the measure of what outputs are possible in a function, in short, what y values are possible from the function. In this graph we have one line going from -1 to 1, and another going from 2,4. Note that there is no line going from 1 to 2. This means there is a break in the range, denoted by a union of the 2 ranges, shown as a U. Therefore the range of this function would look as follows; (-1,1)U(2,4), since there is no line at 3 it cannot be included in the range.