01Xincreasinggraphs

toc = Strictly Increasing & Decreasing Graphs =

Strictly Increasing Graphs
A graph (y = f(x)) is said to be ** strictly increasing ** if
 * b > a** implies that
 * f(b) > f(a)** for the entire domain of f(x)

This means that for all points in the domain, if we step to the right, the y-value must have increased.

This can include stationary points.

** Example 1 **
In the graph on the right (figure 1),

y = f(x) is a ** strictly increasing ** graph over the entire domain, x Î R. {including at x = 1 (stationary point)}

y = g(x) is a ** strictly increasing ** graph in the domain: math x \in \big( -\infty, \; 2 \big] math {including at x = 2 (stationary point)}

Note

If a graph is strictly increasing (or decreasing) then its inverse function is defined.

There is no requirement that the graph be differentiable or continuous.

Strictly increasing (and decreasing) graphs are described as ** one-to-one functions **. {for any x value, there is only one y value and for any y value, there is only one x value}

** Example 2 **
In the graph on the right (figure 2)

y = f(x) is a ** strictly increasing **graph over the entire domain x Î R {including at x = 1 where it is not differentiable (not smooth)}

y = g(x) is a ** strictly increasing **graph over the entire domain x Î R {including at x = 2 where it is not differentiable (not continuous)}

Strictly Decreasing Graphs
In the same way, A graph (y = f(x)) is said to be ** strictly decreasing ** if
 * b > a** implies that
 * f(b) < f(a)** for the entire domain of f(x)

The graph of y = (x² – 1)² (shown to the right) is strictly decreasing in two sections: math \\ \big\{ x:x \leqslant -1 \big\} \\ \\ \textit{and} \\ \\ \big\{ x: 0 \leqslant x \leqslant 1 \big\} math
 * Example 3 **

Note: The graph is __**not**__ strictly decreasing in the combined domain: math \big\{ x:x \leqslant -1 \big\} \cup \big\{ x: 0 \leqslant x \leqslant 1 \big\} math because in this domain, there __are__ pairs of points where a step to the right does __not__ result in a decrease. (eg between x = –1 and x = 0) or (eg between x = –1.2 and x = 0.2)

(Also: it is __not__ a one-to-one graph in this domain.)

We could describe a combined domain which does produce a strictly decreasing graph. For example: math \big\{ x:x < -\sqrt{2} \big\} \cup \big\{ x: 0 \leqslant x \leqslant 1 \big\} math

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