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 of $$f\left( x \right)$$. 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.