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Monotonicity and boundedness

A function $$f$$ is bounded if, for some constant $$M$$ and all $$x\in D_f$$, $$$\vert f(x) \vert \le M.$$$ It is bounded below if $$f(x)\ge -M$$ and bounded above if $$f(x)\le M$$.

The graph of a bounded function lies within two horizontal lines. The graph of a function bounded below remains above an horizontal line.

$$\sin(x)$$ is bounded; $$x^2$$ is bounded below; $$x^3$$ is unbounded.

A function is increasing if, whenever $$x\gt y$$, $$$f(x)\ge f(y).$$$ It is decreasing if $$f(x)\le f(y)$$. A function is strictly increasing or strictly decreasing if the inequalities are all strict. A function is monotonic if it is either always increasing or always decreasing.

The graph of an increasing function "climbs up or stays flat" as $$x$$ increases.

$$x^3$$ is strictly increasing; $$e^{-x}$$ is strictly decreasing; $$\max(x,0)$$ is increasing, but not strictly increasing when $$x\lt 0$$.