Finding the Domain and Range of a Relation from Its GraphThe concepts of domain and range...

Finding the Domain and Range of a Relation from Its Graph

The concepts of domain and range are an important and fundamental part of working with relations and functions. In this chapter, we learned to determine the domain of any relation from its graph using a “vertical boundary line,” and the range by using a “horizontal boundary line.” These approaches to finding the domain and range can be combined into a single step by envisioning a rectangle drawn around or about the graph. If the entire graph can be “bounded” within the rectangle, the domain and range can be based on the rectangle’s related length and width. If it’s impossible to bound the graph in a particular direction, the related x- or y-values continue infinitely. Consider the graph in Figure 1.60. This is the graph of an ellipse (Section 8.2), and a rectangle that bounds the graph in all directions is shown in Figure 1.61.

Figure 1.60

Figure 1.61

The rectangle extends from x = − 3 to x = 9 in the horizontal direction, and from y = 1 to y = 7 in the vertical direction. The domain of this relation is x ∈ [ − 3, 9] and the range is y ∈ [1, 7].

The graph in Figure 1.62 is a parabola, and no matter how large we draw the rectangle, an infinite extension of the graph will extend beyond its boundaries in the left and right directions, and in the upward direction (Figure 1.63).

The domain of this relation is x ∈ ( − ∞, ∞) and the range is y ∈ [−6, ∞).

Finally, the graph in Figure 1.64 is the graph of a square root function, and a rectangle can be drawn that bounds the graph below and to the left, but not above or to the right (Figure 1.65).

The domain of this relation is x ∈ [−7, ∞) and the range is y ∈ [−5, ∞).

Use this approach to find the domain and range of the following relations and functions.