# 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.

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

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.