Geometric Properties of Graphs

The graph in Figure 1 (a) is always rising as we move to the right, a geometric indication that the function f is increasing; that is, as x increases, so does the value f\left( x \right). On the other hand, the graph in Figure 1 (b) is always falling as we move to the right, indicating that the function g is decreasing; that is, as x increases, the value of g\left( x \right) decreases. In Figure 1 (c), the graph doesn’t rise or fall, indicating that h is a constant function whose values h\left( x \right) do not change as we increase x. The following definition makes these ideas more precise.


geometric-graph01
Increasing, Decreasing and Constant Function:

Let the interval I be contained in the domain of the function f.

(i) f is increasing on I if for every two numbers a and b in I with a < b we have f\left( a \right) < f\left( b \right).

(ii) f is decreasing on I if for every two numbers a and b in I with a < b we have f\left( a \right) > f\left( b \right).

(iii) f is constant on I if for every two numbers a and b in I we have f\left( a \right) = f\left( b \right).


geometric-graph02

As Figure 2 illustrates, the domain and range of a function are easily found from its graph. The domain of a function is the set of all abscissas of points on its graph, and the range is the set of all ordinates of points on its graph.