Download Relations & Functions

Relations & Functions
Section 2-1
Definitions
• A relation is a description of the association
between two sets of values.
• The set of input values is called the domain
and the set of output values is called the
range.
• For example, there is an association
(hopefully!) between the color of a traffic light
and the behavior of a driver approaching it.
Traffic Light
Color
Behavior
Red
Stop
Yellow
Slow down
Green
Speed up
Maintain speed
School Schedule
Period
Class
1
Algebra 2
2
Gym
3
Chemistry
4
Study
5
Geometry
6
Cell Phone Direction Pad
Direction
Up
Down
Left
Right
Action
Calculator
Inbox
Pictures
Ringtones
Multiplication
Number
x2
2
4
-1
-2
0.5
1
-1.3
-2.6
0
0
1
2
School Schedule (again!)
Subject
Period
Math
1
Science
2
Gym
3
Study
4
5
6
Functions
• A function is a relation in which each input
value maps to exactly one output value.
• Which of the previous examples are
functions?
Ordered Pairs
• When the input and output values are
numbers, as in the multiplication example, we
can think of the input and output values as x
and y, and represent the relation as a
collection of ordered pairs:
{(2, 4), (-1, -2), (0.5, 1), (-1.3, -2.6), (0, 0), (1, 2)}
• We can also graph these points!
Ordered Pairs (cont’d)
x
y
2
4
-1
-2
0.5
1
-1.3
-2.6
0
0
1
2
Vertical Line Test
• When we graph a relation, we can use the
vertical line test to determine whether or not
it is a function.
• In order to be a function, the graph must have
the property that any vertical line drawn
through it only touches it once. This
corresponds to each input (x) value having
only one output (y) value.
Vertical Line Test (cont’d)
Here’s a problem…
• What if we wanted to expand the previous
example to include more inputs and outputs,
but following the same rule?
• We could write out some more ordered pairs:
…(4, 8), (5, 10), (6, 12), (7, 14)…
… but these are just a few! There are infinitely
many possible ordered pairs that we could
add to that relation.
… and a solution!
• We can represent the relation using the
equation that describes the relationship
between the inputs and outputs:
y = 2x
Solution (cont’d)
• Now if want to know what output value is
produced by the input value 27, we just plug 27
in for x:
y = 2(27)
y = 54
• Similarly, if want to know what input value gives
an output value of -13, plug -13 in for y:
-13 = 2x
x = -6.5
Example 1
• A relation is defined by the equation:
y = x2 + 3
• What are some ordered pairs that are part of
this relation?
(-2, 7), (-1, 4), (0, 3), (1, 4), (2, 7), (3, 12)
• Is this relation a function?
• Yes! For each input (x), there is exactly one
output (y) – to find it, just square and add 3!
Example 2
• A relation is defined by the equation:
|y| = x – 1
• What are some ordered pairs that are a part of
the relation?
{(4, 3), (8, -7), (5, 4), (5, -4)}
• Is this relation a function?
No! The input value 5 has two different output
values: 4 and -4
Domain and Range
• Recall that the domain is the set of input
values, and the range is the set of output
values.
• When a relation is given as an equation, the
domain and range are often difficult to figure
out.
• We need to think about all the possible values
of x and y in the equation.
Example 1
y = x2 + 3
• Given a number as input (x), is there always an
output value for it?
• Yes – just square it and add 3.
• The domain of this relation is all real numbers.
Example 1 (cont’d)
•
•
•
•
y = x2 + 3
Given a number as output (y), can we always find
an input (x) to go with it?
No! For example, try y = -1:
-1 = x2 + 3 has no solution!
In fact, we know that x2 ≥ 0 always, so the output,
which is equal to x2 + 3, satisfies:
x2 + 3 ≥ 3
The range of this relation is {y | y ≥ 3}
Example 2
•
•
•
•
|y| = x – 1
Given a number as input (x), is there always an
output value for it?
No! For example, try x = -5.
|y|= -5 – 1 has no solution!
In order to have a solution, we need:
x – 1 ≥ 0, or solving, x ≥ 1
The domain of this relation is {x | x ≥ 1}
Example 2 (cont’d)
|y| = x – 1
• Given a number as output (y), can we always
find an input (x) to go with it?
• Yes – we can always solve for x.
• The range of this relation is all real numbers.
Function Notation
• Recall that when a relation is a function, there
is exactly one output value for each input
value. With functions, we sometimes use
function notation to represent the output
Replaces y
value:
f(x) = x2 + 3
Function
name
Input
value
Rule for finding the
output value
Function Notation (cont’d)
f(x) = x2 + 3
• f(x) is read “f of x” and refers to the output value
when x is the input value.
• f(-5) refers to the output value when 5 is used as
an input value.
f(-5) = (-5)2 + 3 = 28
• f(-5) does not mean to multiply the variable f by
the number -5!
More examples
f(x) = x2 + 3
f(a) = a2 + 3
f(2a) = (2a)2 + 3 = 4a2 + 3
f(b + 5) = (b + 5)2 + 3
= (b+5)(b+5) + 3 = b2 + 10b + 25 + 3
= b2 + 10b + 28