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Concept

# Increasing and Decreasing Intervals

A function is said to be increasing when, as the $x$-values increase (from left to right), the values of $f(x)$ increase. Conversely, a function is said to be decreasing when, as $x$ increases, $f(x)$ decreases. The graph below shows increasing intervals with green arrows and decreasing intervals with red arrows.

The function above contains two increasing intervals and one decreasing interval. To describe each, use the $x$-values. Commonly a point where a function has a relative maximum or a relative minimum is neither included in an increasing nor a decreasing interval. It is necessary to scan the graph from left to right. \begin{aligned} \text{From left side of graph to} \ x=\text{-} 2 & \quad \rightarrow && \text{increasing} \\ \text{From} \ x=\text{-} 2 \ \text{to} \ x=0 & \quad \rightarrow && \text{decreasing} \\ \text{From} \ x=0 \ \text{to right side of graph} & \quad \rightarrow && \text{increasing} \\ \end{aligned} Although the entire graph cannot be shown, it is reasonable to assume it continues in the same manner. Thus, for all $x$-values less than $x=\text{-} 2,$ $f$ will be increasing. Additionally, for all $x$-values greater than $x=0,$ $f$ will be increasing. Thus, the above intervals can be expressed as follows.

\begin{aligned} \textbf{Increasing interval:} & \ \text{-} \infty < x < \text{-} 2 \ \text{and} \ 0 < x < + \infty \\[0.8em] \textbf{Decreasing interval:} & \qquad \qquad \ \text{-} 2 < x < 0 \\ \end{aligned}