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Topic: monotonic function
Replies: 14   Last Post: Nov 29, 2012 6:53 PM

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Posts: 12,067
Registered: 7/15/05
Re: monotonic function
Posted: Nov 27, 2012 6:08 PM
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Michael Stemper wrote:
>Sepidehshm writes:
>>Thank you for your time.
>>But as I mentioned above, I know the meaning of monotonic
>>unctions. I need examples of these functions.

>How about f(x) = mx + b, with b<>0? That's an example of a
>monotonic function.

Surely you meant to require m<>0 rather than b<>0.

to the OP:

Any increasing function is monotonic.

Any decreasing function is monotonic.

Any monotonic function is either increasing or decreasing.

For any increasing function f(x), the function -f(x)
is decreasing and similarly, for any decreasing function
f(x) the function -f(x) is increasing.

Every monotonic function is one-to-one, hence has an
inverse (defined on the range of the given function).

The inverse of an increasing function is increasing.

The inverse of an decreasing function is decreasing.

The sum of two increasing functions is increasing.

An increasing function plus a constant function is increasing.

An increasing function times a positive constant is increasing.

More generally, an increasing function plus a non-decreasing
function is increasing.

If an increasing function has only positive values
its reciprocal is decreasing.

If two increasing functions have only positive values
their product is increasing.

For any odd positive integer n, the function f(x) = x^n
is increasing.

For any constant b > 1, the function f(x) = b^x is increasing.

Here's just one example of a function which whose monotonicity
follows directly from the above properties.

f(x) = 3^x + 2*(x^5) + sqrt(x) + x*(ln(x)) - 8

It should be obvious how to use those properties to create lots
of other examples.


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