# 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.

__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 $$f\left( a \right) = f\left( b \right)$$.