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Introduction to Points, Vectors, And Functions - At A Glance:

Bounded and Unbounded Functions: To Infinity and Beyond!

You may have met some bounded functions in algebra. Unfortunately, much like a friend of a friend of your ex-girlfriend or ex-boyfriend, you haven't seen them in a while and can't remember their name.

Here is a quick review of the ideas, first.

A function is bounded below if we can find some y-value K that the function never goes below.

In symbols, for all x we have K ≤ f(x).

A function is bounded above if we can find some y-value M that the function never gets above.

In symbols, for all x we have f(x) ≤ M.

If a function isn't bounded, we say it's unbounded.

A function which is unbounded above and increasing grows without bound:

If a function is unbounded below and decreasing, its magnitude grows without bound:

Sample Problems

It's moving day on Calculus Street, and we need to decide which functions to pack up and take with us and which ones to leave behind for the next tenants.

The function f(x) = sin x is bounded above by 1 and below by -1.

We can put this function in our box and put it in the moving truck.

The function f(x) = ex not bounded above, but it is bounded below by 0. It doesn't ever equal 0, but we can still fit it in a box. We can't close the lid, we will have to put it at the top of the pile.

The function f(x) = x3 is not bounded above or below. We will leave this one here for the next tenants.

Be Careful: When a function has an upper and/or lower bound, the bound(s) are y-values, not x-values.

Exercise 1

Exercises. Determine if each function is (a) bounded above (b) bounded below.

  • y = cos x

Exercise 2

Exercises. Determine if each function is (a) bounded above (b) bounded below.

  • f(x) = 5

Exercise 3

Exercises. Determine if each function is (a) bounded above (b) bounded below.

  • f(x) = x2

Exercise 4

Exercises. Determine if each function is (a) bounded above (b) bounded below.

  • y = x

Exercise 5

Exercises. Determine if each function is (a) bounded above (b) bounded below.

Exercise 6

Exercises. Determine if each function is (a) bounded above (b) bounded below.

Exercise 7

Exercises. Determine if each function is (a) bounded above (b) bounded below.

Exercise 8

Exercises. Determine if each function is (a) bounded above (b) bounded below.

Exercise 9

Exercises. Determine if each function is (a) bounded above (b) bounded below.

Exercise 10

Exercises. Determine if each statement is true or false. 

  • If a function is bounded above it must also be bounded below.

Exercise 11

Exercises. Determine if each statement is 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.

Exercise 12

Exercises. Determine if each statement is true or false. 

  • If there is some K such that f(x) ≥ K for all x, then the function f(x) is bounded below. 
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