What does it mean when the intervals of a function are increasing or decreasing? For example,w hat does it mean when it says a function is decreasing on the interval $(-\infty, 0)$ or for example or increasing on the interval $(0,-\infty)$?
Function having increasing and decreasing intervals
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0We don't say that the intervals of a function are increasing/decreasing. We say that a function is increasing/decreasing over an interval. And it means that as you move to the right on the interval, the value of the function increases or decreases. So, if you think of a function just like an airplane that is moving in a time interval, if the airplane is increasing its latitude, it's like the function is increasing and when the airplane is decreasing its altitude, its like the function is decreasing. – 2013-12-04
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A function $f$ is increasing on an interval $I$ if whenever $x$ and $y$ are points in the interval, if $x$ is larger than $y$, then $f(x)$ is larger than $f(y)$. In other words, $f$ is increasing on an interval $I$ if for all $x, y \in I$ with $x > y$, then $f(x) > f(y)$.
Likewise, a function $f$ is decreasing on an interval $I$ if for all $x, y \in I$ with $x > y$, then $f(x) < f(y)$.
See here for more info.
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0Note that this defines what is often called *strictly* increasing. Many take "increasing" to mean *weakly* increasing. – 2013-12-04