Boundedness of a Function

[AquaAngel]

When a function (such as y=2^x ) has an asymptote, is it bounded on the bottom?

 

[stapel]

When a function (such as y=2^x ) has an asymptote, is it bounded on the bottom?

What is the definition (from your book or class notes) of being "bounded"? :wink:

 

[AquaAngel]

"A function f is bounded above if there is some number B that is greater than or equal to every number in the range of f. Any such number B is called an upper bound of f."

The definition is the same for "bounded below", except b is less than or equal to every number in f.

Thanks for the quick reply!

 

[stapel]

"A function f is bounded above if there is some number B that is greater than or equal to every number in the range of f. Any such number B is called an upper bound of f."

The definition is the same for "bounded below", except b is less than or equal to every number in f.

Great! Now, comparing the definition with the behavior of the function, is the function \(\displaystyle f(x)\, =\, 2^x\) bounded below by, say, the value b = 0? :wink:

 

[AquaAngel]

Ok thanks!

 

[HallsofIvy]

Notice that, once you have said "it is bounded below by 0", it is automatically also bound below by any negative number. It is, for example, bounded below by -1 because f(x)> 0 > -1 so f(x)> -1.​