Handout: Functions and Associated Notation
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Reeve Garrett
Sets
In math, sets and functions are probably the most important tools available to model real life
phenomena and communicate ideas. First, let’s give some definitions.
Definition 1.1 A set is a collection of objects, which we refer to as elements.
Generally, we give sets names (oftentimes, we’ll use capital letters), sometimes with special symbols.
Some sets which we’ll encounter frequently in the course and the associated symbols that denote
them are the following:
1. N, the set of natural numbers 1, 2, 3, etc.
2. Z, the set of all integers
3. Q, the set of all rational numbers (fractions of integers)
4. R, the set of all real numbers
5. C, the set of all complex numbers (don’t worry if you don’t know what this means yet)
Notation 1.2 Given a set A and an element a, to express “a is an element of A,” we write a ∈ A.
Similarly, for “a is not an element of A,” we write a ∈
/ A.
Definition 1.3 A subset is a smaller set contained inside another set. For instance, if A and
B are sets, and every element of A is an element of B, we say A is a subset of B, and write
A ⊆ B.
Definition 1.4 Given two sets A and B, the Cartesian product of A and B, denoted A × B
(read “A cross B”), is the set of ordered pairs (a, b), where a ∈ A and b ∈ B.
Notation 1.5 It’s very common to write R × R as R2 . We’ll be doing that.
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Subsets of R
When specifying subsets in R, we generally use set-builder notation. What’s meant by this is
when we mean to say “this set A is all real numbers where condition P holds,” we write
A = {x | P (X) holds}, which we read as “A is the set of all real numbers x such that condition P
holds for x.”
Caution 2.1 In theory, I could replace x with whatever letter or symbol I want (heck, a smiley
face or a giraffe would work just the same!), so long as the symbol replacing x in the definition of
a set via set-builder notation (and similarly for functions, as we’ll see) replaces x everywhere. It
will still mean the same exact thing. Don’t get tripped up when you see different letters in place of
x in homework and exam problems!
Example 2.2 For the set α of all real numbers that are integers when squared, we write
α = {t | t2 ∈ Z}, where α and t can be replaced by pretty much any letters or symbols we want!
Other common subsets of R we’ll deal with are of the forms (where a ∈ R and b ∈ R):
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Handout: Functions and Associated Notation
Reeve Garrett
1. [a, b] = {x | a ≤ x ≤ b}
2. (a, b) = {x | a < x < b}
3. [a, b) = {x | a ≤ x < b}
4. (a, b] = {x | a < x ≤ b}
5. (a, ∞) = {x | x > a}
6. [a, ∞) = {x | x ≥ a}
7. (−∞, b) = {x | x < b}
8. (−∞, b] = {x | x ≤ b}
The moral to this notation is that parentheses mean to exclude the endpoint(s) and square brackets
tell you to include the endpoint(s).
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Subsets of R2 and their graphical representations
Of course, subsets of R aren’t the only sets we care about. Oftentimes, we care about sets in R2 ,
the set of all ordered pairs of real numbers . Like how we can display subsets of R in pictures on
a number line, we can display subsets of R2 as pictures in the coordinate plane. For examples,
see pages 83 and 84 of your textbook.
Notation 3.1 When a subset of R2 is specified by one equality or inequality, instead of writing
the set in set-builder notation, we generally just write the equality or inequality and understand it
to mean all ordered pairs (x, y) satisfying it, and when a problem tells you to graph an equality or
inequality, it’s telling you to graph this set.
Example 3.2 y ≤ x2 + 3x + 2 means {(x, y) | y ≤ x2 + 3x + 2}, which we’ll learn how to represent
graphically as the parabola y = x2 + 3x + 2 with the area below it shaded in, as we’ll learn in detail
later in the course.
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Functions and their graphs
Now, we’re finally ready to talk about functions and graphs.
Definition 4.1 A function is a rule assigning each element in one set to exactly one element of
another set.
Notation 4.2 We generally write f : A → B to mean f is a function assigning each element of
A to an element of B. We call A the domain of f and B the codomain of f . We generally
write f (x) for the value in B that f assigns an element x ∈ A (as cautioned before, there is nothing
special about using the letter x here: it’s just an arbitrarily named placeholder, and it can be replaced
with any other letter or symbol we want!). The range of f is the set {y ∈ B | y = f (x) for some
x ∈ A}. We can make the codomain as big as we want (for instance, we could let the codomain be
,Q or R for a function taking values in Z), so long as it contains the range, so we only really care
about the domain and range of a function.
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Handout: Functions and Associated Notation
Reeve Garrett
Notation 4.3 For a function f that assigns each real number to some algebraic expression of
2
), we write
that number (for instance, when a function f assigns every real number a to √aa4+3
+1
f as f (x) = that algebraic expression with x plugged in (continuing with that example, we would
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write f (x) = √xx4+3
. We understand this to mean that every real number x is assigned to the
+1
expression evaluated at the real number x (for instance, continuing with this example, 3 is assigned
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to f (3) = √334+3
= √1282 , and similarly for every other real number).
+1
Example 4.4 For the function f (the letter f here is arbitrary) which assigns every real number x
to its absolute value |x|, we write f (x) = |x| to represent f . This describes the rule completely and
concisely! Every real number has an absolute value, so the domain of f is all of R. The absolute
value of a number is always greater than or equal to 0, so the range of f is [0, ∞).
Generally, the most useful way to think about functions is through their graphs.
Definition 4.5 Given a function g : α → W , where α and W are sets, the graph of g is the set
of ordered pairs {(a, g(a)) | a ∈ α} ⊆ α × W (I’ve picked the letters g, alpha, and W to again
demonstrate the arbitrariness of naming).
It’s an important fact that graphs describe functions completely, so in a sense, a function and its
graph are the same thing!
In this course, we’ll be almost exclusively looking at functions f : R → R with graphs {(x, y) ∈
R2 | y = f (x)} which we can represent pictorially in the coordinate plane (through extrapolation
from a few important points, of course!).
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Concluding Remark/Reminder
Please supplement this handout by reading all of Sections 2.1 and 2.2 and your notes taken in
lecture. This is a supplement to the textbook’s coverage, not a substitute. I’ve put some content
from Chapter 1 in this handout as well, and you should read the chapter if you’re baffled by any
of what I’ve discussed in the first few sections of this handout. I’ve written this to ensure all of us
are on the same page as far as notation goes - if we don’t use the same notation and terminology,
communicating with each other can be very confusing!
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