1.1 Functions
and Models
Today’s Topics:
• Function Notation
• Domain & Range
• Recognizing Functions
Definition
Function: A function is a rule or
correspondence that assigns to each
element of one set exactly one element of
a second set.
The function may be defined by a set of
ordered pairs, a table, a graph an equation
or a verbal description.
Part I – Function Notation
Ex 1 – Function Notation with Equations
Evaluate the following expressions given the functions
below:
𝟏𝟐
𝐱
g(x) = -3x + 1
f(x) = x2 + 7
𝐡 𝐱 =
a. g(10) =
b. f(3) =
c. h(–2) =
d. f(-7) =
e. h(4) =
f. g(-5) =
Ex 2: Function Notation with a
Graph
y
f(x)
Find:
5
a.
x
-5
5
b.
c.
d.
-5
f(–4) =
f(0) =
f(3) =
f(-5) =
Ex 3: Mortgage
Mortgage A couple can afford $800
per month to purchase a home. As
indicated in the table, if they can
get an interest rate of 7.5%, the
number of years that it will take to
pay off the mortgage is a function of
the dollar amount A of the
mortgage for the home they
purchase.
Amount A
($)
t (years)
40,000
5
69,000
10
89,000
15
103,000
20
120,000
30
a) If the couple wishes to finance $103,000, for how long must
they make payments? Write this in function form if t = f(A).
b) What if f(120,000)? Write a sentence that explains its
meaning.
c) What is f(3∙40,000)?
d) What value of A makes f(A) = 5 true?
e) Does f(3∙40,000) = 3 ∙ f(40,000)? Explain your reasoning.
Part II – Domain & Range
Definition
Domain: The domain of a function is the set
of all possible input values (usually x), which
allows the function formula to work
Definition
Range: The range is the set of all possible
output values (usually y), which result from
using the function formula.
Ex 1: Write the domain and
range of each function.
D:
D:
R:
R:
Ex. 2: Domain and Range

a)
For each of the following functions, determine the
domain. Determine the range of each function.
y = 𝑥2 + 5
1
b)
𝑓(𝑥) = 2 + 2𝑥−5
c)
𝑔(𝑥) = 2 7 − 3𝑥
d)
Determine the domain and range of this function.
𝑥 2 + 𝑦 2 = 25
1
Part III – Recognizing Functions
Recall that a function is a rule or
correspondence that determines exactly
one output for each input.
Ex. 1: Determine whether or not the
indicated relationship represents a
function.
A.
B.
X
Y
0
-100
5
109
10
480
15
505
20
480
25
109
30
-100
.
Does the equation 𝑦 2 = 5𝑥 − 5 define y as a
function of x?
Recognizing Functions
Definition
Vertical Line Test: A set of points in a
coordinate plane is a the graph of a
function if and only if no vertical line
intersects the graph more than one point.
Examples on the white board.
Assignment
Pg. 18-23
#3-17 odd
#26
#28
#29-33 all
#49
#66
Ex. 1: Determine whether or not the
indicated relationship represents a
function.
X
Y
A. 0.
-100
5
109
10
480
15
505
20
480
25
109
30
-100
YES
X
Recall that a function is a rule or
-3
correspondence that determines
exactly one output for each input. 5
8
Y
-1
19
4
25
5
11
5
5
9
0
-10
NO