Determine if the function is (a) bounded above or (b) bounded below.
y = cos x
Answer
Even though the function y = cos x goes up and down like a zebra jumping on a trampoline, it's bounded above by 1 and below by -1:
Example 2
Determine if the function is (a) bounded above or (b) bounded below.
f(x) = 5
Answer
Another constant function? Yep. The function f(x) = 5 is bounded above and below by 5.
Example 3
Determine if the function is (a) bounded above (b) bounded below.
f(x) = x2
Answer
The function f(x) = x2 is unbounded above. It is bounded below by 0, though.
Example 4
Determine if the function is (a) bounded above or (b) bounded below.
y = x
Answer
The function y = x is unbounded above and unbounded below.
Example 5
Determine if the function is (a) bounded above or (b) bounded below.
Answer
The function is bounded above by 0 and unbounded below:
Example 6
Determine if the function is (a) bounded above or (b) bounded below.
Answer
This function is not bounded above and is not bounded below.
Example 7
Is the function is (a) bounded above or (b) bounded below?
Answer
This function is bounded above only.
Example 8
Is the following function (a) bounded above or (b) bounded below?
Answer
This function is bounded both above and below. We can see bounds on the graph:
Example 9
Is the following function (a) bounded above or (b) bounded below?
Answer
This function is bounded below but not bounded above.
Example 10
Is the following statement true or false?
If a function is bounded above it must also be bounded below.
Answer
False. This is like saying just because something has a ceiling, it must have a floor. An example is the function f(x) = -x2, which is bounded above by y = 0 but unbounded below.
Example 11
Is the following statement true or false?
If a function f(x) has an upper bound of M there must be some value of x for which f(x) = M.
Answer
False. The function is bounded above by y = 0, but there is no value of x for which
Example 12
Is the following statement true or false?
If there is some K such that f(x) ≥ K for all x, then the function f(x) is bounded below.
Answer
True. This is the definition of a function that is bounded below.
is bounded above by y = 0, but there is no value of x for which
