Increasing, Decreasing, and Constant Intervals

To determine whether a function is increasing, decreasing, or constant over an interval, ask "What does the y do as the x goes from left to right?". If the y goes up, the function is increasing. If the y goes down, the function is decreasing, and if the y stays flat, the function is constant.

Formally,

Increasing, Decreasing, and Constant Functions

Suppose the a function f is defined over an open interval I and $LaTeX: x_1\:$ $x_1\:$ and $LaTeX: x_2\:$ $x_2\:$ are in the interval I.

$LaTeX: f\left(x\right)\:$ $f\left(x\right)\:$ increases over I if, whenever $LaTeX: x_1<x_2,\:f\left(x_1\right)<f\left(x_2\right).$ $x_1<x_2,\:f\left(x_1\right)<f\left(x_2\right).$
$LaTeX: f\left(x\right)\:$ $f\left(x\right)\:$ decreases over I if, whenever $LaTeX: x_1<x_2,\:f\left(x_1\right)>f\left(x_2\right).$ $x_1<x_2,\:f\left(x_1\right)>f\left(x_2\right).$
$LaTeX: f\left(x\right)\:$ $f\left(x\right)\:$ is constant over I if, for every $LaTeX: x_1\:$ $x_1\:$ and $LaTeX: x_2,\:f\left(x_1\right)=f\left(x_2\right)$ $x_2,\:f\left(x_1\right)=f\left(x_2\right)$ .

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