Relation
Def: A relation is any set of ordered pairs.
Ex) { ( 2, 3), ( 4, -3), ( 5, 9), ( 3, 2) }
Def: The Domain (Input) of a relation is the
set of all x values. D = { 2, 4, 5, 3 }
Def: The Range (Output) of a relation is the
set of all y values. R = { 3, - 3, 9, 2 }
Function
Def: A Function is a relation in which each
value in the domain gives ( Corresponds
to) only one value in the range.
Ex) { ( 3, 2), ( 5, 4), ( 9, 11), ( 5, 20) }
Not a Function.
x = 5 corresponds to y = 4 and y = 20.
Determine if the relation is a
function.
Ex) { ( 2, 3), ( 4, -3), ( 5, 9), ( 3, 2) }
Yes, the relation is a function.
Ex) { ( -4,5), ( 7, 11), ( 0, 1), ( 3, 5)}
Yes, the relation is a function.
Ex) { ( 3,0), (-1, 7), ( 6, 11), (-1, 2)}
No, the relation is not a function.
Domain: { -1, 3, 6} Range: { 0, 2,7, 11}
Functions can be represented
using:
1) Relation or Set of ordered pairs.
{ ( 1, 1), ( 2, 4), (3, 9), (0, 0 ), ( -1,1), ( 2,4),(-3, 9) }
2) Graph
3) Equations
Not all sets of ordered pairs, graphs or equations
are functions. To be a function, the set of
ordered pairs, graph or equation must satisfy
the definition given earlier.
Functions as Equations
{ ( 1, 1), ( 2, 4), ( 3, 9), ( 4, 16)}
Can you write an equation that describes the
relationship between the x and y values?
y = x²
If x = 5, then y = 5² = 25.
An ordered pair was created. ( 5, 25)
If x = -7, then y = ( -7)² = 49.
An ordered pair was created ( -7,49)
Functions can have different
relationships between the x and y
Ex )
{ ( 1, 3), ( 2, 5), ( 3, 7), ( 4, 9)}
y = 2x + 1
Ex)
{ ( 7, 4), ( 8, 5), ( 9, 6), ( 10, 7)}
y=x-3
Function Notation
We now use a new symbol to represent y,
the range value.
We introduce the symbol y = f(x)
The symbol is read “ f of x”
f(x) means the y value that is created by
using a specific x value as input.
Let’s use f(x) = 2x + 1
The x inside the parentheses is the domain
value.
The symbol f(x) is the range value.
The expression 2x + 1 tells us what we need
to do to the domain value, in order to get
our range value.
The entire symbol 2x + 1, represents a
______ value.
Evaluate each expression below
Using f(x) = x³ , evaluate
a)
f( 4)
Using f(x) = -3x – 2, evaluate
a)
f( -3)
b) f( t )
Using f(x) = x², evaluate
a)
f(x – 3)
b) f( x – h)
Using f(x) = x² - 2x, evaluate
a) f( x – h )
b) f( x )
c) f( x – h ) – f(x)
Does the Equation represent y as a
function of x ?
Let’s look at the equation x = y²
If this equation represents a function, then if we
plug in a value for x, then it should produce only
one value for y.
Let x = 4. This gives 4 = y² and so
y = 2 or y = -2
No, x = y² does not represent y as a function of x.
More examples
Ex) x² + y² = 25
Ex)
y x
Ex
y  x3
Ex)
x  y²
Evaluating the Difference Quotient
The Difference Quotient for
functions is defined to
be
Evaluate the difference
quotient for
a) f(x) = 3x + 1
b) f(x) = x²
c) f(x) = x² - 2x
d) f(x) = - 4x
f ( x  h)  f ( x )
h
Answers:
a) 3
b) 2x+h
c) 2x + h – 2
d) - 4
Finding Domains for functions
written as Equations
If we have the function { ( 1, 5), ( 3, 7), (-2,0)}, then
Domain = { 1, 3, -2} and Range = { 5, 7, 0}
What would it mean if I asked you to find the
domain for the function
2
f ( x) 
x 3
To find the Domain for a function when it is written
as an Equation, means to find all real x values
that will produce real y values.
Let’s look at
2
f ( x) 
x 3
Let’s plug x values into the function and see what happens?
Consider the x values as Input and the y values as Output.
Ask yourself. What x values produce y values that are real?
These are the values we want to include in the domain.
Ask yourself. What x values produce y values that are not real?
These are the values we want to exclude from the domain.
Write the answer using interval notation.
How do you get Nonreal y values?
1. Dividing a number by zero, produces a
nonreal y value.
For example: 7/0 = not real or undefined
2. Taking the square root of a negative number.
For example:
4 ?
is the same as ?² = - 4
Domains
Find the domain for each function below. Write the
answer using interval notation.
2x
f(x) = 3x + 2
f ( x) 
x4
x 1
f ( x) 
( x  2)( x  4)
f ( x)  x  3
f ( x)  x
Graphs of Nonfunctions
The relation is not a function.
{ ( 1,2), ( 3, -2), ( -1, 4), ( 1, -2)}
Let’s plot these points on the coordinate
system. What do we notice?
The two points (1,2) and ( 1, -2) have the
same x value but different y value. They
lie on the same vertical line.
Graphs of Functions
The relation is a function
{ (-3, 2), ( -2, 1), ( -1, -3), ( 0, 2), ( 1, 3),
(2,0), ( 3, -2) }
Let’s plot these points on a coordinate
system. What do we notice?
This graph represents a function because
you cannot find two points that lie on the
same vertical line.
Determine which graphs represent
functions.
YES
NO
YES
NO
Find the Domain and Range
Domain [ -5, 4 ]
Range [ 1, 7]
Find the Domain and Range
Domain [ -4 , 3]
Range [ -3 , 0 ]
Find Domain and Range
Domain
Range
[5, )
[0, )
Find Domain and Range
Domain
Range
(, )
(,1]
f(x) is called an
even function. The
graph is symmetric
with respect to the
y –axis.
Notice:
f(2) = f( -2) = 3
f(4) = f( -4) = 1
Lastly, to be an even function means that
f (x) = f( - x )
This function is an
odd function. This
means that the
graph is symmetric
with respect to the
origin.
Notice:
f(2) = 3 and f( -2) = -3
f(5) = 2 and f( -5) = -2
In other words,
f( -2) = - f( 2) = - ( 3) = -3
f(- 5) = - f( 5) = -( 2) = -2
If a function is odd, this means
f(-x) = - f(x)
An x- intercept is a point where a graph crosses the x- axis.
All x – intercepts are points with y = 0
( - 3, 0) and ( 2, 0)
A y – intercept is a point where the graph
crosses the y – axis.
All y – intercepts are points with x = 0.
( 0, 4)
Intervals of Increase or Decrease
We say that f(x) is increasing on the interval ( -6, 0).
We say f(x) is decreasing on the interval ( 0, 4).
When determining intervals of increase or decrease, we read the
graph from left to right. In other words, let the x values get larger.
Over which intervals(months)
was the account increasing?
Jan, Feb, March
Over which
intervals(months) was the
account decreasing?
April, May, July and August
Over which month(s) was the
account constant?
June
Find interval(s) where f(x) is
increasing, decreasing or
constant.
a) f(x) is increasing on
(-5, -1) or ( 4, 7)
b) f(x) is decreasing on
( -1, 4)
c) f(x) is never constant
Determine the intervals where f(x) is increasing, decreasing or
constant.
Increasing on ( -
 , -5) or ( 5, 
Constant on ( -5, 5)
f(x) is never decreasing
)
The average rate of the change as x goes from o months to 5 months,
says that if my money were to change by the same amount every
month, it would change by $80 per month, in order to go from the
original $100 to the final $500. It is the slope the secant line.
Piece-wise Functions
3x  1 for x  0
f ( x)  

  2 x for x  0 
Find a) f( 3)
b) f(0)
c) f(-5)
d) f( 4)
Cell Phone Bill
Let’s say I have a program where I pay $50
if I talk between 0 and 100 minutes,
inclusively.
If I talk pass 100 minutes, they charge 7
cents per minute, plus the $50 for the first
100 minutes.
Let x = number of minutes we talk on the
phone for the month.