Is this sequence and/or function monotone?

Question

I know that from what I am learning, Monotone is where a sequence and/or function is neither increasing or decreasing at a consistent rate.

Take for example, the function: $$P(t): 2+\frac{10}{2t}$$

Is this function monotone or decreasing? I feel as if this function is decreasing at all times according to the math, but at a certain point the function just flatlines. What is the correct answer to this?

Answer 1

• A function is increasing if $f(x)\le f(y)$ whenever $x<y$.

• A function is decreasing if $f(x)\ge f(y)$ whenever $x<y$.

• A function is monotone if it is increasing or decreasing.

Some authors use $f(x)<f(y)$ and $f(x)>f(y)$ in the above definitions.

Any function which is increasing at a constant rate is thus monotone. The function $P(t)=2+\dfrac5t$ is decreasing on $(0,\infty)$ because, if $0<x<y$, then $\dfrac1y<\dfrac1x$, which means $2+\dfrac1y<2+\dfrac1x$, so $P(y)<P(x)$.