Math 8 — Functions and Modeling
Neil Donaldson
Spring 2025
Introduction
This course aims to refresh and reinforce the conceptual foundations behind several topics commonly
encountered in grade-school mathematics. The job of a teacher is often one of selection: choosing
examples and explanations suited to the level and experience of your students. To select effectively,
and to anticipate student questions, your must understand concepts at a higher level than you’ll
likely ever teach. Not all of our topics are central to the grade-school curriculum, and it is not our
goal to teach you how to teach, though the ideas and approaches we’ll explore are often suitable for
a grade-school audience. The mathematics in this course shouldn’t present much difficulty for math
majors, requiring at most elementary calculus and a tiny bit of linear algebra; you should instead be
considering how to explain the material, particularly to students with less mathematical knowledge
than yourself.
We start with two motivational problems.
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1. You wish to travel across the surface of a cube between two oppo-
site vertices so that your path is as short as possible.
Should you follow the path indicated?
If yes, explain why.
If not, how should you find the shortest path?
2. Two houses are to be connected to the elec-
tricity supply using a single connection.
How should we determine where to place
the connection so as to minimize the required
length of wire?
What information do you need in order to
find the connection point?
connection
wires
electric supply
House 1
House 2
Your goal shouldn’t only be to find the right answer! Consider how you might discuss these problems
with grade-school students of different ability levels. Why might calculus not be a sensible approach?
Are there any similarities between the two problems? Brainstorm some strategies. . .
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We are grateful to materials from UT Austin’s UTeach program for suggesting several of the examples in this course
including these motivational problems.
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1 Sets & Functions
1.1 Basic Definitions
Consider how central functions are to mathematics, and how long you’ve been using them. How
would you define “function” to someone with limited mathematical knowledge? Would you use
words like rule, assign, element, domain, vertical line test, etc.? How helpful are these to your audience?
Examples 1.1. How would you explain the idea that the following do or do not represent functions?
1. y = x
2
2. Mon: fish, Tue: pork, Wed: fajitas, Thur: carbonara, Fri: pizza, Sat: fish, Sun: pizza
3. (3, 5), (2, 6), ( 4, 2), (3, 1).
4. x
2
= y
2
After considering the examples, perhaps you settle on a semi-formal definition:
A function f is rule which assigns to each input x exactly one output f (x)
Is this a useful definition? In what ways is it imprecise? Does the imprecision matter?
Of course the answers to these questions depend on your audience! What ideas do you want to
convey to your students and can you do so without overburdening and intimidating them? To begin
working towards a more complete picture, consider what we might allow to be inputs and outputs.
This requires a small amount of set notation.
Definition 1.2. A set A is a collection of objects, or elements.
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The notation a ∈ A means that a is an
element of A, sometimes read ‘a lies in A.’ Sets are often written upper case and elements lower.
A set B is a subset of a set A, written B ⊆ A, if every element of B is also an
element of A: that is,
b ∈ B =⇒ b ∈ A
The picture illustrates sets A, B and elements a, b for which B ⊆ A, a ∈ A,
b ∈ B and a /∈ B (a does not lie in B).
A
B
a
b
Examples 1.3. 1. Suppose the elements of a set A are the numbers 1, 3, 5, 7 and 9. The simplest way
to write this is using roster notation: we list the elements (in any order) between braces
A = {1, 3, 5, 7, 9}
Subsets are commonly expressed using set-builder notation. For example, here is a subset of A:
B = {a ∈ A : 2 < a < 8}
This is read, “The set of a in A such that a lies strictly between 2 and 8.” In roster notation,
B = {3, 5, 7}. Can you express B in other ways using set-builder notation?
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This is enough for our purposes, though a course in set theory will convince you that this definition has its own
problems. Selection is always at work. . .
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2. We summarize several common sets of numbers using informal combinations of roster and
set-builder notation, all of which should be familiar.
Natural numbers N = {1, 2, 3, 4, . . .}. For instance, 5 ∈ N but −3 ∈ N.
Integers Z = {. . . , −2, −1, 0, 1, 2, 3, . . .}. For instance, −4 ∈ Z but
4
5
∈ Z.
Rational numbers (fractions) Q =
p
q
: p ∈ Z, q ∈ N
. For instance −
6
7
∈ Q; in this case
p = −6 is an integer, and q = 7 a natural number.
Real numbers R. For instance,
√
2 ∈ R. A formal definition is difficult, though we often
informally visualize R as a ruler. Intervals are particularly important subsets, e.g.,
[−4, π) = {x ∈ R : −4 ≤ x < π}
is a half-open interval.
You should also be familiar with the Cartesian plane: R
2
= {(x, y) : x, y ∈ R}. The notation
(3, 4) ∈ R
2
here describes a point in the plane with co-ordinates x = 3, y = 4; don’t confuse
this with the interval ( 3, 4) = {x ∈ R : 3 < x < 4} which is a subset of R!
The subset relationships between these sets are in the order listed:
N ⊆ Z ⊆ Q ⊆ R
You should also have informally encountered the notion of irrationality: for instance,
√
2 and π
are real numbers but not rational numbers.
The reason we need this language when discussing functions is that the inputs and outputs of a
function are elements of sets. Here is a very formal definition of “function.”
Definition 1.4. The Cartesian product of sets A, B is the set of ordered pairs
A × B =
(a, b) : a ∈ A, b ∈ B
A function from A to B is a non-empty subset f ⊆ A ×B which satisfies the vertical line test
For each a ∈ A, there is a unique b ∈ B such that (a, b) ∈ f (∗)
Instead of writing f ⊆ A × B and (a, b) ∈ f , we use the more familiar notation
f : A → B and f (a) = b
To a function f : A → B are associated three useful sets:
• Domain: dom f = A is the set of inputs.
• Codomain: codom f = B is the set of possible outputs.
• Range: range f = {b ∈ B : b = f (a) for some a ∈ A} is the set of realized outputs.
This probably isn’t the definition you should give to 10
th
graders, or even to freshman calculus stu-
dents! But what should you do? How much of this is helpful in a a given context?
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Example (1.1.2 cont.). We revisit our food-based example in this formal setting. To properly view
this as a function f : A → B, we have to carefully label the constituent sets.
A =
Mon, Tue, Wed, Thu, Fri
, B =
carbonara, fajitas, fish, pizza, pork
,
f =
(Mon, fish), (Tue, pork), (Wed, fajitas), (Thu, carbonara),
(Fri, pizza), (Sat, fish), (Sun, pizza)
The domain A should be clear, but we had to make a choice for the codomain B: in this case we chose
it to equal to range. Can you suggest a different choice for B? Try the other examples yourself.
Representing Functions
Functions can be represented in various ways. We illustrate a few in an example.
Example 1.5. We consider the familiar formula/rule f (x) = x
2
in several contexts.
Table This presentation is most helpful when the domain is very small.
The table shows the situation when dom f = {−1, 0, 1, 2, 3} and
range f = {0, 1, 4, 9}
Arrows A pictorial arrow diagram might also be helpful when the do-
main is small.
Graph This is the set of ordered pairs
x, f (x)
: x ∈ dom f
: in the
context of the formal definition (1.4), the graph is the function!
For formulæ whose inputs and outputs are real numbers, two con-
ventions are often observed:
• The domain is implied to be all real numbers for which the
formula makes sense.
• The codomain is taken to be the set of real numbers.
If no other information is provided, we’d assume that the function
defined by the formula f (x) = x
2
has both domain and codomain
the entire set of real numbers: f : R → R.
The range of the function is the set of possible outputs, in this case
range f = {x
2
∈ R : x ∈ R} = [0, ∞)
is the half-open interval of non-negative real numbers.
For ‘calculus’ functions like these, the vertical line test (∗) really in-
volves vertical lines; every vertical line intersects the graph in pre-
cisely one point.
In the picture, the dots are the graph when the domain is the finite
set {−1, 0, 1, 2, 3} (as described in the table/arrow-diagram).
x −1 0 1 2 3
f (x) 1 0 1 4 9
−1
0
1
2
3
0
1
4
9
2
4
6
8
10
y
−2 0 2
x
Can you think of other ways to represent a function? How might you decide which to use?
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Exercises 1.1. 1. Let d represent the cost in millions of dollars to produce n cars, where n is measured
in 1000s. As clearly as you can, explain what is meant by d(25) = 431.
2. A movie theater seats 200 people. For any particular show, the amount of money the theater
takes in is a function of the number of people n in attendance. If a ticket costs $25, describe the
domain and range of the function using set notation.
3. Temperature readings T were recorded every two hours from midnight to noon. Time t was
measured in hours from midnight.
t 0 2 4 6 8 10 12
T (
◦
F) 82 75 74 75 84 90 93
(a) Plot the readings and use them to sketch a rough graph of T as a function of t.
(b) Use your graph to estimate the temperature at 10:30 a.m.
4. State parts 1, 3 and 4 of Example 1.1 using the formal language of Definition 1.4. If you have a
function, state the domain and range and explain how you know you have a function. If you
don’t have a function, explain why not.
(Since insufficient information is provided, there is no single correct answer!)
5. (a) Let A = {1, 3, 5, 7, 9}. Explain in words what is meant by the set
B = {x ∈ A : x
2
> 10}
and state B in roster notation.
(b) Find the set C = {x ∈ N : (x −1)
2
< 16} in roster notation.
(c) Find the Cartesian product B ×C in roster notation. Is it the same as C × B?
6. Suppose that f : {−2, −1, 0, 1, 2} → R is defined by the formula f (x) = x
3
−4x + 1.
Describe f using a table, an arrow diagram and a graph.
7. Find the implied domain and range for the functions defined by each rule:
(a) f (x) =
x
2
−4
x−2
(b) g(x) =
√
x
2
−16x (c) h(x) =
1
x
√
4x −x
2
(What is the largest set of real numbers for which the formula makes sense?)
8. You ask your students to determine the range of the function f defined by the rule f (x) = x
2
with domain the interval [−5, 2]. You obtain various responses, including [25, 4], [4, 25] , and
[−25, 4]. What is going wrong? What is the correct answer, and how would you explain it to
your students?
More generally, if dom f = [a, b] (where a ≤ b), what is range f ?
9. The unit circle is often represented by the implicit equation x
2
+ y
2
= 1.
(a) Draw the circle and explain why the full circle isn’t the graph of a function.
(b) Describe two functions f : [−1, 1] → R and g : [−1, 1] → R whose graphs together
comprise the circle. What are the ranges of each function?
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1.2 Linear Polynomials
Perhaps the simplest functions are the linear polynomials, whose graphs are straight lines,
y = f (x) = mx + c where m, c are constants (∗)
Linear polynomials make very simple models: increase the input by ∆x and the output changes by
∆y = m∆x regardless of the starting value x. Given experimental data or a physical situation relating
two quantities x and y, a linear model is an linear polynomial (∗) relating these variables. In practice,
models are approximations to the real-world data. Later in the course we’ll consider what should be
meant by, and how to find, a ‘good’ linear model for approximately linear data.
Some of your earliest forays into algebra likely involved finding equations of straight lines.
Example 1.6. Find the equation of the straight line through the points A = (1, 3) and B = (4, 1).
Suppose the polynomial is y = mx + c. Since both A and B sat-
isfy this equation, we start by substituting both points into the
equation to find two relationships between m and c
(
3 = m + c
1 = 4m + c
This is a system of two linear equations in two unknowns (m, c).
By now you should know several ways to solve such, but con-
sider what might be easiest for a grade-school student. . .
0
1
2
3
4
y
0 1 2 3 4 5
c
x
m
1
A
B
Regardless of how you phrase it (solve one equation for c and substitute into the other, subtract one
question from the other, etc.), we obtain
−2 = 3m =⇒ m = −
2
3
=⇒ c = 3 − m =
11
3
whence the required polynomial is y =
1
3
(11 − 2x).
As the picture suggests, the gradient/slope m represents how far one climbs/falls on travelling one
unit to the right. The y-intercept c is the intersection of the graph with the vertical axis.
The above process works for any two points A = (x
0
, y
0
) and B = (x
1
, y
1
) provided x
0
= x
1
: is it
clear why this should be the case? The details are in Exercise 5. You might feel that such a problem
is too abstract for your students, that such a ‘proof’ might be too intimidating. Indeed it might be
counterproductive for some students, but consider several counterpoints:
• Once a student has developed comfort with concrete examples as above, Exercise 5 helps sum-
marize and unify what they’ve learned. A general/abstract discussion helps build confidence
by convincing a student that any such problem can be solved the same way.
• The most helpful elementary proofs are those which essentially replicate an example abstractly.
Exercise 5 is not some abstract existence proof—it involves no trickery—it simply reinforces the
core technique by applying it in the most general situation.
• Helping and encouraging students to think abstractly is one of the overarching learning out-
comes of all mathematics. You might get push-back, but it’s part of the job. . .
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Example 1.7. Often the challenge of modeling lies in converting a word problem into algebra—don’t
underestimate how hard students find this! Here is a simple, though disguised, straight line model.
Beaker A contains a 300 ml solution of 2% acid. Beaker B contains 400 ml of acid of unknown con-
centration. The beakers are mixed together to produce an acid with concentration 6%. What was the
concentration in beaker B?
Given your mathematical experience, it should seem natural to denote the unknown concentration
(beaker B) by x. After mixing, we have a 700 ml solution containing 300 ×
2
100
+ 400x ml of pure
acid, whence its concentration is a linear polynomial function of x:
C(x) =
6 + 400x
700
The problem is now easily solved: C(x) =
6
100
⇐⇒ x =
9
100
= 9%.
Parametrized Lines Straight lines admit an alternative visualization. Imagine placing a ruler so that
its zero point is at the origin O = (0, 0) and the “1” lies at a point C = (c
1
, c
2
). If t (a real number) is
the measure on the ruler, then the points on the line have co-ordinates
tC = (tc
1
, tc
2
) (∗)
To describe the line through points A and B, place a ruler so that 0
corresponds to A and 1 to B. Now slide the ruler so that A moves
to the origin O: this amounts to subtracting the co-ordinates of A
from all points on the line. We obtain a parallel line through the
origin, with B transformed to the point C = B − A. Putting this
together with (∗) results in a parametrized description of the line:
(x, y) = A + tC = A + t(B − A) = (1 − t)A + tB
1
2
1
0
3
2
−
1
2
A
B
C = B − A
O
Contrast the parametrized description of a line with the linear polynomial approach: for instance,
one challenge is that a line may be parametrized using infinitely many distinct rulers (choose any
two points on the line!), whereas the linear polynomial description is unique. Does the parametrized
approach have any advantages? Which description is easier to understand or to work with? Which
fits better with your intuitive understanding of line? Which might cause a grade-school student the
greater challenge?
In the Exercises we make sure that the two descriptions of a line correspond. The discussion is little
more than the generalization of an example.
Example 1.8. The line through points A = (3, 6) and B = (−1, 4) may be parametrized by
(x, y) = (1 − t)(3, 6) + t(−1, 4) =
3 −4t, 6 −2t
To convert this to a linear polynomial, first solve for t in terms of x,
x = 3 −4t =⇒ t =
1
4
(3 − x)
before substituting into our expression for y:
y = 6 −2t = 6 −
2
4
(3 − x) =
1
2
x −
9
2
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Exercises 1.2. 1. The cost of gasoline is $4.20 per gallon on January 1
st
and $4.90 on March 1
st
. State
a linear function/model for the cost of gasoline as a function of time.
2. You have a choice of three different cell-phone plans.
(a) No monthly charge and 10¢ per minute for all calls.
(b) $10 per month and 5¢ per minute for all calls.
(c) $30 per month, regardless of how many calls you make.
How should you determine which plan to purchase?
3. Revisit Exercise 1.1.3. Find an approximate linear model T(t) = mt + c for this data.
(There is no perfect answer)
4. Revisit the beakers problem (Example 1.7). This time suppose we know that the concentration
in beaker B is 9%. How much from beaker B should we pour into beaker A to obtain an acid
with concentration 5%? Would you consider this a linear polynomial problem? Why/why not?
5. Suppose points A = (x
0
, y
0
) and B = (x
1
, y
1
) are given.
(a) If x
1
= x
0
, use the method of Example 1.6 to find the equation y = mx + c of the line
through these points.
(b) Now use the parametrized approach where A corresponds to 0 and B to 1. If, in addition,
x
1
= x
0
, make things match up with your answer to part (a).
What parametrization do you get if A = (0, c) and B = (1, m + c)?
(c) Part (a) provides an algebraic justification of the claim made on page 7, that the linear
polynomial description of a line is unique (‘the equation’). How might you help a student
believe this claim if the algebra is unconvincing or too intimidating?
(Think about Example 1.6)
6. A straight line is sometimes described as the set of points (x, y) ∈ R
2
satisfying an equation of
the form
ax + by = c
for some constants a, b, c where a, b are not both zero. How does this approach differ from our
use of linear polynomials?
7. Throughout mathematics (particularly within linear algebra), a function f : R → R is said to be
linear if it satisfies the condition
For all λ, x ∈ R, f (λx) = λ f (x)
Is this the same thing as a linear polynomial? Explain.
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