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Functions (1-7)
Objective: Determine whether
a relation is a function. Find
function values.
Identify Functions


A function is a relationship between input and
output.
In a function, there is exactly one output for each
input.
Key Concept

A function is a relation in which each
element of the domain is paired with
exactly one element of the range.
 Examples
y
Domain
Range
4
-3
5
2
0
3
2
2
4
-1
x
-4
-2
2
-2
-4
4
Example 1

Determine whether each relation is a
function. Explain.
a. Domain
Range
-2
-8
0
0
2
8
4
16
Function. Each element of the
domain is paired with only one
element of the range.
b.
Domain
Range
-7
-4
2
5
-12
-9
3
0
Function. Each element of the
domain is paired with only one
element of the range.
Graphs
A graph that consists of points that are not
connected is a discrete function.
 A function graphed with a line or smooth
curve is a continuous function.

Example 2

There are three lunch periods at a school. During the first period,
352 students eat. During the second period, 304 students eat.
During the third period, 391 students eat.
a.
Make a table showing the number of students for each of the three
lunch periods.
Period
Number of Students
b.
1
2
3
352
304
391
Determine the domain and range of the function.

D = {1, 2, 3}

R = {304, 352, 391}
Example 2

There are three lunch periods at a school. During the first period,
352 students eat. During the second period, 304 students eat.
During the third period, 391 students eat.
c.
Write the data as a set of ordered pairs. Then graph the data.

(1, 352)

(2, 304)

(3, 391)
Example 2

There are three lunch periods at a school. During the first period,
352 students eat. During the second period, 304 students eat.
During the third period, 391 students eat.
d.
State whether the function is discrete or continuous. Explain your
reasoning.

Discrete. The points are not connected.
Graphs




You can use the vertical line test to see if a graph
represents a function.
If a vertical line intersects the graph more than once,
then the graph is not a function.
Otherwise, the relation is a function.
One way to perform the vertical line test is to use a
pencil.

Place your pencil vertically on the graph and move from
left to right.
 If the pencil passes over the graph in only one place, then
the graph represents a function.
Graphs
y
y
y
4
4
4
3
3
3
2
2
2
1
1
1
x
-4
-3
-2
-1
-1
1
2
3
4
5
x
-4
-3
-2
-1
-1
1
2
3
4
5
x
-4
-3
-2
-1
-1
-2
-2
-2
-3
-3
-3
-4
-4
-4
-5
-5
-5
Function
Not a Function
1
Function
2
3
4
5
Example 3

Determine whether x = -2 represents a
function.
y
X
Y
-2
-1
-2
0
-2
1
-2
2
4
3
Not a Function
2
1
x
-4
-3
-2
-1
-1
-2
-3
-4
-5
1
2
3
4
5
Find Function Values


Equations that are functions can be written in a form called
function notation.
For example, consider y = 3x – 8.
Equation
y = 3x – 8




Function Notation
f(x) = 3x – 8
*f(x) is read “f of x”.
In a function, x represents the elements of the domain, and
f(x) represents the elements of the range.
Suppose you want to find the value in the range that
corresponds to the element 5 in the domain.
This is written f(5) and is read “f of 5.”
The value f(5) is found by substituting 5 for x in the equation.
Example 4

For f(x) = 3x – 4, find each value.
a.
f(4) = 3(4) – 4
= 12 – 4
=8
b.
f(-5) = 3(-5) – 4
= -15 – 4
= -19
f(4) = 8
f(-5) = -19
Example 5

If h(t) = 1248 – 160t + 16t2, find each value.
h(3) = 1248 – 160(3) + 16(3)2
= 1248 – 160(3) + 16(9)
h(3) = 912
= 1248 – 480 + 144
= 768 + 144
b. h(2z) = 1248 – 160(2z) + 16(2z)2
= 1248 – 160(2z) + 16(4z2)
= 1248 – 320z + 64z2
a.
h(2z) = 1248 – 320z + 64z2
Check Your Progress

Choose the best answer for the
following.
A.
X
Y
3
-2
4
6
5
2
-1
3
Is this relation a function? Explain.
A.
B.
C.
D.
Yes; for each element of the domain, there is
only one corresponding element in the range.
Yes; it can represented by a mapping.
No; it has negative x-values.
No; both -2 and 2 are in the range.
Check Your Progress

Choose the best answer for the
following.
A.
X
3
1
2
3
Is this relation a function? Explain.
A.
B.
C.
D.
No; the element 3 in the domain is paired with
both 2 and -1 in the range.
No; there are negative values in the range.
Yes; it is a line when graphed.
Yes; it can be represented in a chart.
Y
2
-2
-4
-1
Check Your Progress

Choose the best answer for the following.

At a car dealership, a salesman worked for three days. On the
first day, he sold 5 cars. On the second day he sold 3 cars. On
the third he sold 8 cars. Make a table showing the number of
cars sold for each day.
A.
B.
C.
D.
.
.
.
.
Day
1
2
3
Numbers of Cars Sold
8
3
5
Day
5
3
8
Numbers of Cars Sold
1
2
3
Day
1
2
3
Numbers of Cars Sold
2
2
2
Day
1
2
3
Numbers of Cars Sold
5
3
8
Check Your Progress

Choose the best answer for the following.
 Determine
A.
B.
C.
whether 3x + 2y = 12 is a function.
Yes
No
Not enough information
6
y
4
2
x
-6
-4
-2
2
-2
-4
-6
4
6
Check Your Progress

Choose the best answer for the following.
A.
If f(x) = 2x + 5, find f(3).
A.
B.
C.
D.
8
7
6
11
f(3) = 2(3) + 5
=6+5
Check Your Progress

Choose the best answer for the following.
B.
If f(x) = 2x + 5, find f(-8).
A.
B.
C.
D.
-3
-11
21
-16
f(-8) = 2(-8) + 5
= -16 + 5
Check Your Progress

Choose the best answer for the following.
function h(t) = 180 – 16t2 represents the
height of a ball thrown from a cliff that is 180
feet above the ground.
 The
A.
Find h(2).
A.
B.
C.
D.
164 ft
116 ft
180 ft
16 ft
h(2) = 180 – 16(2)2
= 180 – 16(4)
= 180 – 64
Check Your Progress

Choose the best answer for the following.
function h(t) = 180 – 16t2 represents the
height of a ball thrown from a cliff that is 180
feet above the ground.
 The
B.
Find h(3z).
A.
B.
C.
D.
180 – 16z2 ft
180 ft
36 ft
180 – 144z2 ft
h(3z) = 180 – 16(3z)2
= 180 – 16(9z2)