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Relation
A relation is a correspondence
between two sets where each element
in the first set, called the domain, corresponds to at least one element in the
second set, called the range.
Relation
Person Blood type
ORDERED PAIR
Michael A (Michael, A)
Tania A (Tania, A)
Dylan AB (Dylan, AB)
Trevor 0 (Trevor, O)
Megan 0 (Megan, O)
Relation
Person Blood type
ORDERED PAIR
Michael A (Michael, A)
Tania A (Tania, A)
Dylan AB (Dylan, AB)
Trevor 0 (Trevor, O)
Megan 0 (Megan, O)
Relation
Person Blood type
ORDERED PAIR
Michael A (Michael, A)
Tania A (Tania, A)
Dylan AB (Dylan, AB)
Trevor 0 (Trevor, O)
Megan 0 (Megan, O)
The domain is the set of all the first components.
{Michael, Tania, Dylan, Trevor, Megan}
The range is the set of all the second components.
{A, AB, O}
Relation
Person Blood type
ORDERED PAIR
Michael A (Michael, A)
Tania A (Tania, A)
Dylan AB (Dylan, AB)
Trevor 0 (Trevor, O)
Megan 0 (Megan, O)
The domain is the set of all the first components.
{Michael, Tania, Dylan, Trevor, Megan}
The range is the set of all the second components.
{A, AB, O}
Function
Function
• A function is a correspondence between two sets where each element in the first set, called the domain, corresponds to exactly one element in the second set, called the range
Function
• A function is a correspondence between two sets where each element in the first set, called the domain, corresponds to exactly one element in the second set, called the range
• Note that the definition of a function is more restrictive than the definition of a relation.
Function
Time of day Competition
1:00 P.M. Football
2:00 P.M. Volleyball
7:00 P.M. Soccer
7:00 P.M. Basketball
Functions Defined by Equations
Functions Defined by Equations
y = x2 − 3x
Functions Defined by Equations
y = x2 − 3x
x y = x2 − 3x y
Functions Defined by Equations
y = x2 − 3x
x y = x2 − 3x y
1
Functions Defined by Equations
y = x2 − 3x
x y = x2 − 3x y
1 y = (1)2 − 3(1)
Functions Defined by Equations
y = x2 − 3x
x y = x2 − 3x y
1 y = (1)2 − 3(1) −2
Functions Defined by Equations
y = x2 − 3x
x y = x2 − 3x y
1 y = (1)2 − 3(1) −2
5
Functions Defined by Equations
y = x2 − 3x
x y = x2 − 3x y
1 y = (1)2 − 3(1) −2
5 y = (5)2 − 3(5)
Functions Defined by Equations
y = x2 − 3x
x y = x2 − 3x y
1 y = (1)2 − 3(1) −2
5 y = (5)2 − 3(5) 10
Functions Defined by Equations
y = x2 − 3x
x y = x2 − 3x y
1 y = (1)2 − 3(1) −2
5 y = (5)2 − 3(5) 10
1.2
Functions Defined by Equations
y = x2 − 3x
x y = x2 − 3x y
1 y = (1)2 − 3(1) −2
5 y = (5)2 − 3(5) 10
1.2 y = (1.2)2 − 3(1.2)
Functions Defined by Equations
y = x2 − 3x
x y = x2 − 3x y
1 y = (1)2 − 3(1) −2
5 y = (5)2 − 3(5) 10
1.2 y = (1.2)2 − 3(1.2) −2.16
Functions Defined by Equations
y = x2 − 3x
x y = x2 − 3x y
• Since the variable y depends on what value of x is selected, we denote y as
the dependent variable. (output)
Functions Defined by Equations
y = x2 − 3x
x y = x2 − 3x y
• Since the variable y depends on what value of x is selected, we denote y as
the dependent variable. (output)
• The variable x can be any number in the domain; therefore, we denote x as the independent variable. (input)
Function Notation
Function Notation
• The notation y = f(x) denotes that the variable y is function of x.
Function Notation
• The notation y = f(x) denotes that the variable y is function of x.
INPUT FUNCTION OUTPUT EQUATION
x f f (x) f (x) = 2x + 5
Function Notation
• A Linear function is a function defined by an equation that can be written in the form
f(x) = mx + b , or y = mx + b
where m is the slope of the line graph and
(0, b) is the y - intercept
Function Notation
• A Linear function is a function defined by an equation that can be written in the form
f(x) = mx + b , or y = mx + b
where m is the slope of the line graph and
(0, b) is the y - intercept
Ex. y = -3x + 8 f(x) = 5x – 4
The Graph of the Function
• The graph of the function is the graph of the ordered pairs (x, f(x)), that define the function.
Use the given graphs to evaluate the function.
Find f (0), f (1). f (2) , 4f (3),
Find x such that f (x) = 10, f (x) = 2
Find x such that f (x) = 10, f (x) = 2
Use the given graphs to evaluate the function.
T(−5)
T(−2)
T(4)
Vertical Line Test
• Given the graph of an equation, if any vertical line that can be drawn intersects the graph at no more than one point, the
equation defines a function of x.
This test is called the vertical line test.
Vertical Line Test
Evaluating the Difference Quotient
Evaluating the Difference Quotient
( ) ( )f x h f x
h
Evaluating the Difference Quotient
For the function f (x) = x2 − x, find
( ) ( )f x h f x
h
Falling Objects: Firecrackers.
• A firecracker is launched straight up, and its height is a function of time,
h(t) = −16t2 + 128t, where h is the height in feet and t is the time in seconds with
t = 0 corresponding to the instant it launches. What is the height 4 seconds after launch?