Module 1 Lesson 1 Remediation Notes

Relation

Function

Every equation/graph/set of

Functions are just relations

ordered pairs represents a
relation, but sometimes a
relation is a function.

in which the x values of its
points (ordered pairs) do not
repeat.
If a graph passes the vertical
line test, then it is the graph
of a function.

To determine if a graph is a function,
we use the vertical line test.
If it passes the vertical line test then it
is a function.
If it does not pass the vertical line test
then it is not a function.

Vertical Line Test:
1.Draw a vertical line through the
graph.
2. See how many times the vertical
line intersects the graph at any one
location.
If Only Once – Pass (function)
If More than Once – Fail (not
function)

Is this graph a function?

Only
crosses at
one point.
Yes, this is a function because it passes the vertical line test.

Is this graph a function?

Crosses at more than
one point.

No, this is not a function because it does not pass the vertical line test.

To determine if a table represents a
function, we look at the x column
(domain).
If each number in the x column
appears only once in that column, it
is a function.

Relations and Functions
You can use the vertical line test to determine whether a relation is a function.

Vertical Line Test
If no vertical line intersects a
graph in more than one point,
the graph represents a function.

If some vertical line intercepts a
graph in two or more points, the
graph does not represent a function.

Relations and Functions
State the domain and range of the relation shown
in the graph. Is the relation a function?

y

(-4,3)

(2,3)

The relation is:

{ (-4, 3), (-1, 2), (0, -4), (2, 3), (3, -3) }
The domain is:

{ -4, -1, 0, 2, 3 }

x

(-1,-2)

The range is:

{ -4, -3, -2, 3 }

(0,-4)

Each member of the domain is paired with exactly one member of the range,
so this relation is a function.

(3,-3)

Is this relation a function?

X
1
2
3
4

Y
5
6
5
8

Every number just
appears once.

Yes, this is a function because each number in the x column only
appears once.

Is this relation a function?

X
24
6
10
10

Y
7
9
8
10

The number 10
appears more than
once.

No, this is not a function because 10 appears in the x column more than
once.

To Evaluate a Function for f(#):
Plug the # given in the (#)

into all x’s
Simplify

Try these…
http://www.mathslideshow.com/Alg2/Lesson2-1/fv4.htm

Functions
Remember f(x), g(x), h(x), … all just mean y.

We use f(x), g(x), h(x), … when we have more than

one y = equation.

Review
Evaluate

f ( x) = x 2 − 2 x + 5

for

f (3).

f(3) = (3)2 – 2(3) + 5
f(3) = 8

Evaluate

f ( x) = 5 x 3 − 2 x − 8

for

f (−1).

f(-1) = 5(-1)3 – 2(-1) – 8
f(-1) = -11

Basic function operations
Sum

( f + g) ( x) = f ( x) + g ( x)

Difference

( f – g) ( x) = f ( x) – g ( x)

Product

( f × g )( x ) =

Quotient

f ( x)
f 
, g ( x) ≠ 0
( f g) ( x) =  ( x) =
g ( x)
g

©1999 by Design Science, Inc.

15

f ( x) × g ( x)

f ( x) = 2 x + 3

g ( x) = −5 x − 9

f ( x) − g ( x) = (2 x + 3) − (−5 x − 9)
You MUST DISTRIBUTE
the NEGATIVE

f ( x) − g ( x) = 2 x + 3 + 5 x + 9
f ( x) − g ( x) = 7 x + 12

g ( x) = −5 x − 9

f ( x) = 2 x + 3

f ( x) • g ( x) = ( 2 x + 3)( − 5 x − 9 )
You MUST

f ( x) • g ( x) = −10 x 2 − 18 x − 15 x − 27
f ( x) • g ( x) = −10 x − 33 x − 27
2

FOIL

Domain and Range:
If you are given a set of ordered pairs or a graph

(which you would find the ordered pairs all by
yourself)
The x values are the DOMAIN
The y values are the RANGE

{ (-3,5) , (-1, 6), (0, 4), (2, 3.5), (6, 13), (6, 29}

Domain: { -3, -1, 0, 2, 6 }

Range: { 3.5, 4, 5, 6, 13, 29}

Domain and Range:
If the equation is a line (y = mx + b or y = #)
DOMAIN AND RANGE ARE ALL REAL NUMBERS

ALWAYS!!!!!!!!!!!!

!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

Domain and Range:
If there is an x in the denominator of a fraction, you

need to find the value of x that makes the ENTIRE
DENOMINATOR equal zero.
This number is the EXCEPTION to the DOMAIN of
all real numbers.
could be anything
x

Domain is all real numbers except 0

could be anything
x −5

Domain is all real numbers except 5

could be anything
x+9

Domain is all real numbers except -9

Domain and Range:
If you are given a line segment
The DOMAIN (x values) is written like

#< x < #
The RANGE (y values) is written like
#< y < #

#< x < #
#< y < #

Domain and Range:
If you are given a parabola
The DOMAIN is ALWAYS ALL REAL

NUMBERS
The RANGE (y values) is written like
y > #
or
y< #

Find domain and range from an equation
Most of the functions you study in this course will have all real numbers for both the domain
and range. We’ll only look at the domain for exceptions:
1. Fractions: cannot have the denominator (bottom) = 0, so domain cannot be any
x-value that makes the denominator= 0
Examples
3
f ( x) =
x

y=

Domain: x≠0

f ( x) =

x
x−3

Domain: x≠3 (it’s okay for x=0 on top!)

x2 +1
x 2 − 1 Domain: x≠1 or -1 because they both make the denominator=0

Question: How can you calculate which values make the denominator = 0? Set up the
equation denominator = 0 and solve it. Those values are NOT allowed!

Examples
y

5

●

●
●

-5
-5

y

5

5

Domain: {-3,-2,1,3}
Range: {0, -3}
*Don’t repeat y

Domain: y ≥ 0
Range:
*Graph continues rt

x

x

x

●
x≥3

y

5

5
5

5

Domain: − 2 ≤ x ≤ 1
Range: y=4 or {4}

*x is between -2 and 1

Domain: x is any real #
Range: y ≤ 2
*Graph continues down

5
5

Domain: {x| −

5

4≤ x≤ 3}

Range: {y| − 4 ≤ y ≤ 3 }
*This is “set notation”

Domain: x is any real #
Range: y is any real #
*Graph continues all ways

Does the graph represent a function?
Name the domain and range.
x

y

x

y

Yes
D: all reals
R: all reals
Yes
D: all reals
R: y ≥ -6

Does the graph represent a function?
Name the domain and range.
x

No
D: x ≥ 1/2
R: all reals

y

x

y

No
D: all reals
R: all reals

Visit these sites for remediation:
http://www.purplemath.com/modules/fcnops.htm
http://www.wtamu.edu/academic/anns/mps/math/mathlab/col_algebra/col
_alg_tut30b_operations.htm
http://teachers.henrico.k12.va.us/math/hcpsalgebra2/2-1.htm

Module 1 Lesson 1 Remediation Notes

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Module 1 Lesson 1 Remediation Notes

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