Download B-1 Relations and Functions

Linear Relations and Functions
B-1
Relations and Functions
ACT WARM-UP


Emily scored 145, 170, and 165 in 3 bowling
games. What should she score on her next
bowling game if she wants to have an
average score of exactly 170 for the 4
games?
A) 200 B) 195
C) 185
D) 175 E 165
145+170+165+𝑥

4
= 170; Solve: 480 + x = 680;
x = 200; Therefore, A) 200 is the score
needed on her next bowling game.
Objectives



Identify the domain and range of relations
and functions
Determine whether a relation is a function
Find functional values
Essential Question
How can you determine whether a
relation is a function through
1.Ordered pairs,
2.Mapping diagrams,
3.Tables,
4.Graphs,
5.Words?
A relation is a pairing of input values with
output values. It can be shown as a set of
ordered pairs ( x, y ), where x is an input and
y is an output.
The set of input values for a relation is called
the domain, and the set of output values is
called the range. The graph of a relation is
the set of points in the coordinate plane
corresponding to the ordered pairs in the
relation. An x-coordinate is sometimes called
an abscissa and a y-coordinate is sometimes
called an ordinate.
A function is a special type of relation in
which each element of the domain is paired
with exactly one element of the range. If any
input of a relation has more than one output,
the relation is not a function. A mapping
shows how each member of the domain is
paired with each member of the range.
-3
0
2
1
2
4
One-to-one function
-1
1
4
Function, not
one-to-one
3
5
-3
1
5
0
1
6
Not a function
State the domain and range
of the relation shown in the
graph. Is the relation a
function?
The relation is {(1, 2), (3, 3),
(0, –2), (–4, 0), (–3, 1)}.
Answer: The domain is
{–4, –3, 0, 1, 3}. The range
is {–2, 0, 1, 2, 3}. Each member
of the domain is paired with exactly one member of the
range, so this relation is a function.
A function whose graph is made up of unconnected points is
called a discrete function.
State the domain and range
of the relation shown in the
graph. Is the relation a
function?
Answer: The domain is
{–3, 0, 2, 3}. The range
is {–2, –1, 0, 1}. Yes, the
relation is a function.
Transportation The table
shows the average fuel
efficiency in miles per
gallon for light trucks for
several years. Determine
whether it represents a
function.
Answer: Yes, there is
only one fuel efficiency
rating (output) for each
year (input).
Year
Fuel Efficiency
(mi/gal)
1995
1996
20.5
20.8
1997
1998
20.6
20.9
1999
2000
20.5
20.5
2001
20.4
Health The table shows
the average weight of a
baby for several months
during the first year.
Determine whether it
represents a function.
Answer: Yes, the table
represents a function.
There is only one weight
for each age.
Age
(months)
Weight
(pounds)
1
2
12.5
16
4
22
6
9
24
25
12
26
Every point on a vertical line has the same xcoordinate, so a vertical line cannot represent
a function itself, but can be used to
determine whether a relation is a function by
using the vertical line test on a graph. If no
vertical line intersects a graph in more than
one point, the graph represents a function.
When two points on the graph of a relation
are intersected by a vertical line, this means
those two points have the same x value but
different y values. That is, one domain value
is paired with more than one range value.
Graph the relation represented by
Make a table of values to find ordered pairs that satisfy the
equation. Choose values for x and find the corresponding
values for y. Then graph the ordered pairs.
x
y
–1
0
1
2
–4
–1
2
5
(2, 5)
(1, 2)
(0, –1)
(–1, –4)
Find the domain and range.
Since x can be any real number,
there is an infinite number of
ordered pairs that can be
graphed. All of them lie on the
line shown. Notice that every
real number is the x-coordinate
of some point on the line. Also,
every real number is the
y-coordinate of some point
on the line.
(2, 5)
(1, 2)
(0, –1)
(–1, –4)
Answer: The domain and range are both all
real numbers.
Determine whether the relation is a function.
This graph passes the vertical
line test. For each x value, there
is exactly one y value.
(2, 5)
(1, 2)
Answer: Yes, the equation
represents
a function.
(0, –1)
(–1, –4)
Graph the relation represented by
Make a table. In this case, it is easier to choose y values
and then find the corresponding values for x. Then sketch
the graph, connecting the points with a smooth curve.
x
y
5
2
1
2
5
–2
–1
0
1
2
(5, 2)
(2, 1)
(1, 0)
(2, –1)
(5, –2)
Find the domain and range.
Every real number is the
y-coordinate of some point on
the graph, so the range is all real
numbers. But, only real numbers
that are greater than or equal to
1 are x-coordinates of points on
the graph.
Answer: The domain is
The range is all
real numbers.
(5, 2)
(2, 1)
(1, 0)
(2, –1)
(5, –2)
.
Determine whether the relation is a function.
x
y
5
2
1
2
5
–2
–1
0
1
2
(5, 2)
(2, 1)
(1, 0)
(2, –1)
(5, –2)
You can see from the table and the vertical line test that
there are two y values for each x value except x = 1.
Answer: The equation
a function.
does not represent
Example 4d
Determine whether the relation is a function.
From last name to Social Security number.
A last name, such as Smith, from the domain
would be associated with many different
Social Security numbers. The relation from
last name to Social Security number is not a
function.
Some sets of ordered pairs can be described by
using an equation. When the set of ordered pairs
described by an equation satisfies the definition of
a function, the equation can be written in
functional notation. When an equation represents
a function, the variable, usually x, whose values
make up the domain is called the independent
variable. The other variable, usually y, is called the
dependent variable because its values depend on
x. When a function is graphed, the independent
variable is graphed on the horizontal axis and the
dependent variable is graphed on the vertical axis.
The equation y = 2x + 1 can be written
as f (x) = 2x + 1. The symbol f (x)
replaces the y and is read “f of x.” The f is
just the name of the function. It is not a
variable that is multiplied by x.
Suppose you want to find the value in the
range that corresponds to the element 4 in
the domain of the function. This is written
as f (4) and is read “f of 4.” The value f (4)
is found by substituting 4 for each x in the
equation. Therefore, f (4) = 2 (4) + 1 or 9.
Letters other than f can be used to
represent a function.
Given
, find
Original function
Substitute.
Simplify.
Answer:
Given
, find
Original function
Substitute.
Answer:
Given
find each value.
and
a.
Answer: 6
b.
Answer: 0.625
c.
Answer:
Essential Question
How can you determine whether a
relation is a function through
1.Ordered pairs, All of the x’s are different
member of the domain is paired
2.Mapping diagrams, Each
with exactly one member of the range.
3.Tables, All of the x values are different
4.Graphs, Passes the vertical line test
5.Words? Each element of the domain is
associated with exactly one
element in the range.
Math Humor

Why did the relation need a math tutor?

It failed the vertical line test.