If f(x) is strictly increasing then f(x) must also be non-decreasing.
Answer
True or false questions that test knowledge and understanding of mathematics definitions can be tricky. We will take our time and break each one apart.
True. If f(x) is strictly increasing that means f(x) is getting bigger all the time. If f(x) is getting bigger all the time it can't be getting smaller; it must be non-decreasing.
Example 2
Is the following statement true or false?
If f(x) is non-increasing then f(x)must be strictly decreasing.
Answer
False. If f(x) is non-increasing that means f(x) isn't getting bigger anywhere. However, that doesn't necessarily mean f(x) must be getting smaller.
A constant function is an excellent example:
Example 3
Is the following statement true or false?
The function f(x) must be at least one of the following: strictly increasing, non-decreasing, strictly decreasing, or non-increasing.
Answer
False. If the function f(x) changes directions, it need not be any of the above on the entire real line.
Example 4
Is the following function (a) strictly increasing, (b) non-decreasing, (c) strictly decreasing, (d) non-increasing, or (e) none of these? There may be more than one correct answer.
y = 5
Answer
The correct answers are (b) and (d). A trick question already? The constant function y = 5 isn't going anywhere. It's not getting bigger or smaller.
Since the function isn't getting bigger, it can't be strictly increasing.
Since the function isn't getting smaller, it is non-decreasing.
Since the function isn't getting smaller, it can't be strictly decreasing.
Since the function isn't getting bigger, it is non-increasing.
Example 5
Is the following function (a) strictly increasing, (b) non-decreasing, (c) strictly decreasing, (d) non-increasing, or (e) none of these options? There may be more than one correct answer.
y = x2
Answer
The function y = x2 gets smaller on the interval (-∞, 0) and bigger on the interval (0,∞).
Over the whole real line, the function is (e) none of the above. Since it changes directions, this function is not strictly increasing, non-decreasing, strictly decreasing, or non-increasing.
Example 6
Is the following function (a) strictly increasing, (b) non-decreasing, (c) strictly decreasing, (d) non-increasing, or (e) none of these options? There may be more than one correct answer.
y = x3
Answer
The function y = x3 is getting bigger over the whole real line, this function is strictly increasing (a) and non-decreasing (b).
Example 7
Is the following function (a) strictly increasing, (b) non-decreasing, (c) strictly decreasing, (d) non-increasing, or (e) none of the above? There may be more than one correct answer.
y = -x
Answer
The function y = -x is getting smaller over the whole real line. This function is strictly decreasing (c) and non-increasing (d).
Example 8
Consider the function f(x) graphed below.
What is the largest interval on which f(x) is
strictly increasing?
Answer
These types of questions can seem terrifying. We have to select the largest possible intervals each of these different statements apply.
[x1, 0]
Example 9
Consider the function f(x) graphed below.
What is the largest interval on which f(x) is
non-decreasing?
Answer
The interval [x1, x3] is the one we want. Remember that any function that is strictly increasing on an interval is also non-decreasing on that interval.
Example 10
Consider the function f(x) graphed below.
What is the largest interval on which f(x) is
strictly decreasing?
Answer
[x3, x4]
Example 11
Consider the function f(x) graphed below.
What's the largest interval where f(x) is non-increasing?
