Increasing and decreasing functions

A function is said to be increasing when it increases as the variable increases and decreases as the variable decreases. A function is said to be decreasing when it decreases as the variable increases and increases as the variable decreases.

The graph of a function indicates plainly whether it is increasing or decreasing.

Example 8.2.1 (1) For instance, consider the function

whose graph (Figure 8.5) is the locus of the equation

Figure 8.5: SAGE plot of

$\includegraphics[height=4cm,width=4cm]{exp-fcn2.eps}$

As we move along the curve from left to right the curve is rising; that is, as increases the function () always increases. Therefore is an increasing function for all values of .

(2) On the other hand, consider the function whose graph (Figure 8.6) is the locus of the equation .

Figure 8.6: SAGE plot of

$\includegraphics[height=4cm,width=7cm]{cubic-fcn2.eps}$

Now as we move along the curve from left to right the curve is falling; that is, as increases, the function () always decreases. Hence is a decreasing function for all values of .

(3) That a function may be sometimes increasing and sometimes decreasing is shown by the graph (Figure 8.7) of

$\displaystyle y = 2x^3 - 9x^2 + 12x - 3.$

Figure 8.7: SAGE plot of

$\includegraphics[height=4cm,width=7cm]{cubic-fcn3.eps}$

As we move along the curve from left to right the curve rises until we reach the point A when , then it falls from A to the point B when , and to the right of B it is always rising. Hence

(a): from $x = -\infty$ to the function is increasing;
(b): from to the function is decreasing;
(c): from to $x = +\infty$ the function is increasing.

The student should study the curve carefully in order to note the behavior of the function when and . Evidently A and B are turning points. At A the function ceases to increase and commences to decrease; at B, the reverse is true. At A and B the tangent (or curve) is evidently parallel to the -axis, and therefore the slope is zero.