Note: Always give exact answers and always put your
answers in interval notation when applicable.
If the value of x determines the value of y, we
say that “y is a function of x.”
If there is more than one value of y
corresponding to a particular x-value then y
is not determined by x.
◦ (ie, y is NOT a function of y)
Vertical Line Test:
A graph is a graph of a function if and only if
there is no vertical line that passes through
the graph more than once.
Example 1: Does the following graph
represent a function?
Example 2: Does the following graph
represent a function?
Example 3: Do the following relations represent
functions?
(a)
(c)
Fido
Bossy
Silver
Frisky
Polly
450
550
2
40
8
3
(b)
Civil War
1963
WWI
1950
WWII
1939
Korean
1917
Vietnam
1861
x
-1
1
5
7
10
15
y
4
-3
9
4
-3
9
The set of all possible x-values is defined as
the domain.
The set of all resulting y-values is defined as
the range.
*Always write your answers in interval notation
unless instructed otherwise
Example 4: Solve the equation for y to
determine if it represents y as a function of x.
If so, determine the domain and range.
3𝑦 − 3𝑥 2 = 12𝑥 + 9
Example 5: Find the domain and range of
each of the following functions.
(a)
(b)
(c)
Graphs of Relations
and Functions
**A function is 1-1 if and only if it’s graph
passes the vertical line test AND the horizontal
line test.**
Make a table listing ordered pairs that satisfy
the following equation. Then Graph the
equation using the ordered pairs.
2
𝑦 =1−𝑥
X
-2
-1
0
1
2
Y
Graph the following equation. Is it 1-1? What
is the Domain and Range?
𝑦 = 𝑥−3
Determine if the following relations describe
y as a function of x. Graph each relation.
(a) 𝑦 =
16 − 𝑥 2
(b) 𝑦 = − 16 − 𝑥 2
(c) 𝑦 2 = 16 − 𝑥 2
Graph each relation. Determine the intervals
when the following relations are increasing,
decreasing, and constant.
(a)
(b)
(c)
Transformations
There are 2 categories of transformations.
1.
Rigid Transformations
2.
Nonrigid Transformations
There are 3 different rigid transformations:
1. Vertical
◦ Shifts up and down
2. Horizontal
◦ Shifts left and right
3. Reflection
◦ Reflects over an axis

f(x) + a is f(x) shifted upward a units

f(x) – a is f(x) shifted downward a units

f(x + a) is f(x) shifted left a units

f(x – a) is f(x) shifted right a units

–f(x) is f(x) flipped upside down
◦ ("reflected about the x-axis")
How many units is each function shifted? In
which direction?
(a)
(b)
(c)
How many units are each graph shifted? In
which direction.
(a)
(b)
(c)
There are 2 types of nonrigid transformations.
1. Stretching
- Let a > 1. Then y = af(x) stretches the graph
by a factor of a.
2. Shrinking
- Let 0 < a < 1. Then y = af(x) shrinks the
graph by a factor of a.
Graph the following equations on your calculator.
Use transformations to graph the following
function. State the domain and range.
Note: Be sure to follow the order of operations
while translating the function. “Please Excuse My
Dear Aunt Sally.” (Parentheses, exponents,
multiplication/division, addition/subtraction).
Describe the transformation in words. Then
verify by graphing on your calculator.
(a)
(b)
Operations With Functions
Provided that g(x) ≠ 0.
Let
following:
(a) f + g
(b) f – g
(c) f ⋅ g
(d) f/g
. Evaluate the
Let
following:
(a) y - w
(b) y/w
. Evaluate the
If f and g are two functions, the composition
of f and g, written f ∘ g, is defined as follows:
Let
following:
(a) (f∘g)(x)
(b) (g∘f)(x)
. Evaluate the
Inverse Functions
***A function has an inverse if and only if the
function is 1-1.***
The inverse of a one-to-one function f(x) is
the function f -1 such that:
Note: The domain of f(x) is the range of f -1(x)
The range of f(x) is the domain of f -1(x)
To find the inverse of a function f(x):
1)
2)
3)
4)
5)
Replace f(x) with y
Interchange x and y
Solve the equation for y.
Replace y with f -1(x).
Verify that Df = Rf-1 and vice versa.

Find the equation of the inverse of f(x) =2x-3
Graph the inverse of the following function:
𝑓 𝑥 = 𝑥3 + 6
*Remember: reflect the graph of f(x) over the
line y=x to get the graph of the inverse.
Find the inverse of the following function.
x
y
2
0
3
1
6
2

Determine if it’s a function

Graphs of functions

Finding Domain and Range

Operations of Functions

Transformations

Functions and their Inverses