Evgeny.Makarov said:
Alexmahone said:
Amer said:
You can't find a decreasing function whose graph lies outside the band $\{(x,y):|y|\le\varepsilon\}$? If you don't know a precise formula, can you at least describe how such function behaves?Alexmahone said:
Evgeny.Makarov said:
Yes, but I was asking really about f(x). Here are the possible behaviors of f(x).Alexmahone said:
A bounded, decreasing function is one that has a finite range and is always decreasing as its input values increase. This means that the function has both an upper and lower bound, and the output values always decrease as the input values increase.
To determine if a function is bounded and decreasing, you can plot the function on a graph and see if it has a finite range and a downward trend. Alternatively, you can also check the function's derivative, which should always be negative for a bounded, decreasing function.
Knowing if a function is bounded and decreasing is important because it can tell us about the behavior and limits of the function. It can also be useful for optimization problems, as bounded, decreasing functions tend to have a global minimum.
Yes, a function can be bounded and decreasing on an infinite interval. For example, the function $\displaystyle f(x) = \frac{1}{x}$ is bounded and decreasing on the interval $\displaystyle x > 0$.
Some common examples of bounded, decreasing functions include exponential functions such as $\displaystyle f(x) = e^{-x}$, logarithmic functions such as $\displaystyle f(x) = \ln(x)$, and power functions such as $\displaystyle f(x) = x^{-2}$. These functions have a finite range and always decrease as the input values increase.