Algebra 2 A
Chapter 2
Section 1
Relations and Functions
Relations and Functions
ALGEBRA 2 LESSON 2-1
(For help, go to Skills Handbook page 848 and Lesson 1-2.)
Graph each ordered pair on the coordinate plane.
1. (–4, –8)
2. (3, 6)
3. (0, 0)
4. (–1, 3)
5. (–6, 5)
Evaluate each expression for x = –1, 0, 2, and 5.
6. x + 2
7. –2x + 3
8. 2x2 + 1
9. |x – 3|
1
Solutions
1.
4.
6. x + 2 for x = –1, 0, 2, and 5:
–1 + 2 = 1; 0 + 2 = 2; 2 + 2 = 4; 5 + 2 = 7
7. –2x + 3 for x = –1, 0, 2, and 5:
–2(–1) + 3 = 2 + 3 = 5; –2(0) + 3 = 0 + 3 = 3;
–2(2) + 3 = –4 + 3 = –1; –2(5) + 3 = –10 + 3 = –7
2.
5.
8. 2x2 + 1 for x = –1, 0, 2, and 5:
2 • (–1)2 + 1 = 2 • 1 + 1 = 2 + 1 = 3;
2 • 02 + 1 = 2 • 0 + 1 = 0 + 1 = 1;
2 • 22 + 1 = 2 • 4 + 1 = 8 + 1 = 9;
2 • 52 + 1 = 2 • 25 + 1 = 50 + 1 = 51
9. |x – 3| for x = –1, 0, 2, and 5:
|–1 – 3| = |–4| = 4; |0 – 3| = |–3| = 3;
|2 – 3| = |–1| = 1; |5 – 3| = |2| = 2
3.
Relations
 Relation – a set of pairs of input and output
values
Can be written in ordered pairs (x,y)
Can be graphed on a coordinate plane
 Domain – the set of all input values
The x values of the ordered pairs
 Range – the set of all output values
The y values of the ordered pairs
 When writing domains and ranges:
Use braces { }
Do not repeat values NON-Example: {3,3,5,7,9}
2
Graph the relation {(–3, 3), (2, 2), (–2, –2), (0, 4), (1, –2)}.
Try These Problems
a)
{(0,4),(-2,3),(-1,3),(-2,2),(1,-3)}
b)
{(-2,1),(-1,0),(0,1),(1,2)}
3
Write the ordered pairs for the relation. Find the domain and
range.
{(–4, 4), (–3, –2), (–2, 4), (2, –4), (3, 2)}
The domain is {–4, –3, –2, 2, 3}.
The range is {–4, –2, 2, 4}.
Mapping Diagram
 Another way to represent a relation (beside
traditional graphing)
 Links elements of the domain with
corresponding elements of the range
 How To make a mapping diagram:
Make two lists – place numbers from least to greatest
 Domains on the left
 Ranges on the Right
Draw arrows from corresponding domains to ranges (x’s
to y’s)
4
Make a mapping diagram for the relation
{(–1, 7), (1, 3),(1, 7), (–1, 3)}.
Domain
Range
-1
3
1
7
Try These Problems
Make a Mapping Diagram for each relation.
a) {(0,2),(1,3),(2,4)}
b) {(2,8),(-1,5),(0,8),(-1,3),(-2,3)}
0
2
-2
1
3
-1
2
4
0
2
3
5
8
This one is also referred to as one-to-one
function since each element of the range
is paired with exactly one element of the
domain.
5
Functions
 Function – a relation in which each
element of the domain is paired with
EXACTLY one element of the range
 All functions are relations, but not all
relations are functions!!!
There are several ways to determine if a
relation is or is not a function:
 Mapping Diagram
If any element of the domain (left) has more
than one arrow from it
 List of ordered pairs
Look to see if any x values are repeated
 Coordinate Plane
Vertical Line Test – If a vertical line passes
through more than one point on the graph then
the relation is NOT a function.
6
Determine whether each relation is a
function.
a)
b)
-2
0
5
-1
-1
3
0
4
2
3
This is NOT a function
because -2 is paired with both
-1 and 3.
-1
3
5
This is a function because
every element of the
domain is paired with
exactly one element of the
range.
Try These Problems
Determine whether each relation is a function.
a)
b)
2
5
-1
-3
3
6
0
7
4
8
1
10
7
Not a function
Function
7
Vertical Line Test
Try These Problems
Use the Vertical Line Test to determine whether each
graph represents a function.
a)
Not a Function
b)
c)
Function
Not a Function
8
Discrete verse continuous relations
 Discrete- look at the graph the points are
not connected
 Continuous- have an infinite # of elements
and when graphed as a line or curve with
no breaks
Y = -2.5x -7
continuous
Discrete
continuous
Function Rules
 Function Rule – expresses an output value in
terms of an input value
Examples:
y = 2x
f(x) = x + 5
C = πd
 Function Notation –
f(x) is read as “f of x”
This does NOT mean f times x !!!!
f(3) is read as “f of 3”: It means evaluate the
function when x = 3. (plug 3 into the equation)
Any letters may be used C(d)
h(t)
9
Find ƒ(2) , g(2) and h(2).
a. ƒ(x) = –x2 + 1
ƒ(2) = –22 + 1 = –4 + 1 = –3
ƒ(2) = –3
b. g(x) = |3x|
g(2) = |3 • 2| = |6| = 6
c. h(x) = 9
1–x
h(2) = 9
=
1–2
g(2) = 6
9 = –9
–1
h(2) = -9
Try These Problems
Evaluate each function for a domain of {-3, 0, 5}
a)
a)
a)
f(x) = 3x – 5
f(-3) = 3(-3) – 5 = -9 – 5 = -14
f(0) = 3(0) – 5 = 0 – 5 = -5
f(5) = 3(5) – 5 = 15 – 5 = 10
g(a) = ¾ a – 1
g(-3) = ¾ (-3) – 1= -9/4 – 4/4 = -13/4
g(0) = ¾ (0) – 1 = 0 – 1 = -1
g(5) = ¾ (5) – 1 = 15/4 – 4/4 = 11/4
p(y) = -1/5 y + 3/5
p(-3) = -1/5 (-3) + 3/5 = 3/5 + 3/5 = 6/5
p(0) = -1/5 (0) + 3/5= 0 + 3/5 = 3/5
p(5) = -1/5 (5) + 3/5 = -5/5 + 3/5 = -2/5
Your using the
same function, f(x),
3 times to find
different output
values.
10