As we move on the graph of a function from left to right (which corresponds to the increase in the argument $x$) if the graph constantly moves upwards, we say that the function is increasing (Figure 1(a)), and if the graph constantly moves downward, we say that the function is decreasing (Figure 1(b)).
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| (a) Graph of an increasing function | (b) Graph of a decreasing function |
Figure 1
- We say $f(x)$ is increasing on $I$, if for every pair of points $x_{1},x_{2}$ in $I$ satisfying the condition $x_{1}<x_{2}$, we have $f(x_{1})<f(x_{2})$.
- We say $f(x)$ is decreasing on $I$, if for every pair of points $x_{1},x_{2}$ in $I$ satisfying the condition $x_{1}<x_{2}$, we have $f(x_{1})>f(x_{2})$.
- Note that $I$ can be finite (or bounded) or infinite (unbounded).
- An interval on which the function is increasing is called an interval of increase of a function while an interval on which the function is decreasing is called an interval of decrease.
- A function that is either increasing or decreasing on an interval is said to be monotonic on the interval.
For example, $f(x)=x^{2}$ is decreasing on $(-\infty,0]$ and is increasing on $[0,\infty)$ (Figure 2(a)). The function $g(x)=x^{3}$ is increasing on $(-\infty,\infty)$ (Figure 2(b)).
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| (a) Function $f(x)=x^{2} $ is decreasing on $(-\infty,0] $ and is increasing on $[0,\infty) $ . | (b) Function $g(x)=x^{3} $ is increasing in its entire domain $(-\infty,\infty)$ . |
From Figure 3, we note that $g$ is decreasing on $(\infty,0]$ and on $[1,\infty)$; the point of the graph always moves downward as we move from left to right. However, $g$ is not a decreasing function on its entire domain $(-\infty,0]\cup[1,\infty)$ because if $x_{1}=0$ and $x_{2}=1$, although $x_{1}<x_{2}$, we have $g(x_{1})=g(x_{2})=0$. Thus $g$ is not monotonic on $(-\infty,0]\cup[1,\infty)$.
