Transcript Document

Chapter 1
Book cover will go
here
Introduction
to Functions
and Graphs
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Functions and
Their
Representations
Learn function notation
Represent a function four different ways
Define a function formally
Identify the domain and range of a function
Use calculators to represent functions (optional)
Identify functions
Represent functions with diagrams and
equations
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Basic Concepts
The following table lists the approximate distance y
in miles between a person and a bolt of lightning
when there is a time lapse of x seconds between
seeing the lightning and hearing the thunder.
x (seconds)
5
10 15 20 25
y (miles)
1
2
3
4
5
The value of y can be found by dividing the
corresponding value of x by 5.
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Basic Concepts
This table establishes a special type of relationship
between two sets of numbers, where each valid
input x in seconds determines exactly one output y
in miles.
x (seconds)
5
10 15 20 25
y (miles)
1
2
3
4
5
The table represents or defines a function f,
where function f computes the distance between
an observer and a lightning bolt.
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Basic Concepts
The distance y depends on the time x, and so y is
called the dependent variable and x is called the
independent variable.
The notation y = f(x) is used to emphasize that f is a
function (not multiplication). It is read “y equals f of
x” and denotes that function f with input x produces
output y. That is,
f (Input) = Output
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Function Notation
The notation y = f (x) is called function
notation. The input is x, the output is y,
and the name of the function is f.
Name
y = f (x)
Output
Input
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Function Notation
The variable y is called the dependent
variable, and the variable x is called the
independent variable. The expression
f(20) = 4 is read “f of 20 equals 4” and
indicates that f outputs 4 when the input is
20. A function computes exactly one output
for each valid input. The letters f, g, and h
are often used to denote names of
functions.
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Domain and Range of a Function
The set of all meaningful inputs x is called the
DOMAIN of the function.
The set of corresponding outputs y is called the
RANGE of the function.
A function f that computes the height after t
seconds of a ball thrown into the air, has a domain
that might include all the times while the ball is in
flight, and the range would include all heights
attained by the ball.
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Representation of Functions
Functions can be represented by
Verbal descriptions
Tables
Symbols
Graphs
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Verbal Representation (Words)
In the lightning example,
“Divide x seconds by 5 to obtain y miles.”
OR
“f calculates the number of miles from a lightning
bolt when the delay between thunder and lightning
is x seconds.”
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Numerical Representation
(Table of Values)
Here is a table of the lightning example using
different input-output pairs (the same relationship
still exists):
x (seconds) 1
2
3
4
y (miles) 0.2 0.4 0.6 0.8
5
1
6
7
1.2 1.4
Since it is inconvenient or impossible to list all
possible inputs x, we refer to this type of table as a
partial numerical representation.
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Symbolic Representation (Formula)
In the lightning example,
x
f x  
5
Similarly, if a function g computes the square of a
number x, then
g x   x 2
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Graphical Representation (Graph)
A graph visually pairs and x-input with a y-output.
Using the lightning data:
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Graphical Representation (Graph)
The scatterplot suggests a line for the graph of f.
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Formal Definition of a Function
A function is a relation in which each element of
the domain corresponds to exactly one element in
the range.
The ordered pairs for a function can be either
finite or infinite.
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Example: Evaluating a function and
determining its domain
Let a function f be represented symbolically by
x
f x 
.
x 1
(a) Evaluate f(2), f(1), and f(a + 1)
(b) Find the domain of f.
Solution
(a)
2
f 2 
2
2 1
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Example: Evaluating a function and
determining its domain
(b) The expression for f is not defined when the
denominator x – 1 = 0, that is, when x = 1. So the
domain of f is all real numbers except for 1.
a 1
f  a  1 
 a  1  1
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Set-Builder Notation
The expression {x | x ≠ 1} is written in set- builder
notation and represents the set of all real numbers x
such that x does not equal 1.
Another example is {y | 1 < y < 5}, which represents
the set of all real numbers y such that y is greater
than 1 and less than 5.
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Example: Evaluating a function
symbolically and graphically
A function g is given by g(x) = x2 – 2x, and its graph
is shown.
(a) Find the domain and
range of g.
(b) Use g(x) to evaluate
g(–1).
(c) Use the graph of g to
evaluate g(–1).
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Example: Evaluating a function
symbolically and graphically
(a) The domain for g(x) = x2 – 2x, is all real numbers.
(b) g(–1) = (–1)2 – 2(–1) = 1 + 2 = 3
(c) Find x = –1 on the x-axis. Move
upward to the graph of g. Move
across (to the right) to the y-axis.
Read the y-value: g(–1) = 3.
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Example: Find the domain and
range graphically
A graph of
f x   x  2
is shown.
(a) Evaluate f(1)
(b) Find the domain
and range of f.
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Example: Find the domain and
range graphically
(a) Start by finding 1 on the x-axis. Move up and
down on the grid. Note that we do not intersect the
graph of f. Thus f(1) is undefined.
(b) Arrow indicates x and y increase without
reaching a maximum.
Domain is in green:
D = {x | x ≥ 2}
Range is in red:
R = {y | y ≥ 0}
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Identifying Functions: Vertical Line
Test
If every vertical line intersects a graph at no more
than one point, then the graph represents a
function.
Note: If a vertical line intersects a graph more
than once, then the graph does not represents a
function.
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Example: Identifying a function
graphically
Use the vertical line test to determine if the graph
represents a function.
(a)
Solution
(b)
(a) Yes
(b) No
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Functions Represented by Diagrams
and Equations
There are two other ways that we can represent, or
define, a function:
• Diagram
• Equation
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Diagrammatic Representation
(Diagram)
Function
Sometimes referred to as mapping; 1 is the
image of 5; 5 is the preimage of 1.
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Diagrammatic Representation
(Diagram)
Not a function
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Functions Defined by Equations
The equation x + y = 1 defines the function f given
by
f(x) = 1 – x
where y = f(x).
Notice that for each input x, there is exactly one y
output determined by y = 1 – x.
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Example: Identifying a function
Determine if y is a function of x.
a) x = y2
(b) y = x2 – 2
Solution
(a) If we let x = 4, then y could be
either 2 or –2. So, y is not a
function of x.
The graph shows it fails the
vertical line test.
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Example: Identifying a function
(b) y = x2 – 2
Each x-value determines
exactly one y-value, so y is
a function of x.
The graph shows it passes
the vertical line test.
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