Download Chp 1 - Tarleton State University

1
Introduction to
Functions and Graphs
1.1
Numbers, Data, and Problem Solving
1.2
Visualization of Data
1.3
Functions and Their Representations
1.4
Types of Functions and Their Rates of
Change
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1.1
Numbers, Data and
Problem Solving
♦ Recognize common sets of numbers
♦ Learn scientific notation and use it in
applications
♦ Apply problem solving strategies
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Natural Numbers and Integers
• Natural Numbers (or counting numbers)
are numbers in the set N = {1, 2, 3, ...}.
• Integers are numbers in the set
I = {… 3, 2, 1, 0, 1, 2, 3, ...}.
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Rational Numbers
• Rational Numbers are real numbers which can be
expressed as the ratio of two integers p/q where q  0
Note that:
• Every integer is a rational number.
• Rational numbers can be expressed as decimals
which either terminate (end) or repeat a sequence
of digits.
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Irrational Numbers
• Irrational Numbers are real numbers which are not
rational numbers. Irrational numbers
• Cannot be expressed as the ratio of two integers.
• Have a decimal representation which does not
terminate and does not repeat a sequence of digits.
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Example of Classifying Real Numbers
• Classify each number as one or more of the following:
natural number, integer, rational number, irrational
number.
25 ,
3
8,
3.14,
22
.01010101...,
,  11
7
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Scientific Notation
• A real number r is in scientific notation
when r is written as c x 10n, where 1  c  10
and n is an integer.
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Examples of Evaluating Expressions
Involving Scientific Notation
Example 1
Evaluate (5 x 106) (3 x 104), writing the
result in scientific notation and in standard form.
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Example 2
5  106
,
4
writing the answer in scientific
2  10
Evaluate
notation and in standard form.
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1.2
Visualization of
Data
Learn to analyze one-variable data
♦ Find the domain and range of a relation
♦ Graph a relation in the xy-plane
♦ Calculate the distance between two points
♦ Find the midpoint of a line segment
♦ Learn to graph equations with a calculator
(optional)
♦
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Two-Variable Data: Relations
• A relation is a set of ordered pairs.
• If we denote the ordered pairs by (x, y)
• The set of all x  values is the DOMAIN.
• The set of all y  values is the RANGE.
Example
• The relation {(1, 2), (2, 3), (4, 4), (1, 2), (3,0), (0, 3)}
• Has domain D =
• And range R =
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The relation {(1, 2), (2, 3), (4, 4), (1, 2), (3, 0), (0, 3)}
has graph
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Distance Formula
The distance d between two points
(x1, y1) and (x2, y2) in the xy-plane is
d  ( x2  x1 ) 2  ( y2  y1 ) 2
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Example of Using Distance Formula
Use the distance formula
to find the distance
between the two points
(2, 4) and (1, 3).
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Midpoint Formula
The midpoint of the
segment with endpoints
(x1, y1) and (x2, y2)
in the xy-plane is
 x1  x2 y1  y2 
,


2 
 2
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Example of Using Midpoint Formula
Use the midpoint formula
to find the midpoint of the
segment with endpoints
(2, 4) and (1, 3).
Midpoint is:
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1.3
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
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Idea Behind a Function
• Recall that a relation is a set of ordered
pairs_______.
• If we think of values of x as being ________
and values of y as being _________, a
function is a relation such that
• for each input there is __________output.
This is symbolized by
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Function Notation
• y = f(x)
• Is pronounced “y is a function of x.”
• Means that given a value of x (input), there is
exactly one corresponding value of y (output).
• x is called the ______________variable as it
represents inputs, and y is called the
___________ variable as it represents outputs.
• Note that: f(x) is NOT f multiplied by x. f is
NOT a variable, but the name of a function (the
name of a relationship between variables).
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Domain and Range of a Function
• The set of all meaningful inputs is called
the DOMAIN of the function.
• The set of corresponding outputs is
called the RANGE of the function.
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.
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Slide 1- 20
Example 1
• Suppose a car travels at 70 miles per hour.
Let y be the distance the car travels in x
hours.
Then y = 70 x.
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Example 2
Given the following data, is y a function of x?
Input x
Output y
3
6
4
6
8
5
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Example 3
• Undergraduate Classification at Tarleton
State University (TSU) is a function of
Hours Earned. We can write this in
function notation as C = f(H).
• Why is C a function of H?
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C = f(H)
• Classification of
Students at SHU
From Catalogue
No student may be classified
as a sophomore until after
earning at least 30 semester
hours.
No student may be classified
as a junior until after
earning at least 60 hours.
• Evaluate f(20)
• Evaluate f(30)
• Evaluate f(0)
• Evaluate f(61)
No student may be classified
as a senior until after
earning at least 90 hours.
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What is the domain of f?
What is the range of f?
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Questions: Identifying Functions
• Referring to the previous example concerning TSU, is
hours earned a function of classification? That is, is H =
f(C)? Explain why or why not.
• Is classification a function of years spent at TSU? Why or
why not?
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Answers
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• Given x = y2, is y a function of x?
• Given x = y2, is x a function of y?
• Given y = x2 2, is y a function of x?
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Five Ways to Represent a Function
(page 31)
•
•
•
•
•
Verbally
Numerically
Diagrammaticly
Symbolically
Graphically
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C = f(H)
(Referring to previous TSU example)
• Verbal Representation.
• If you have less than 30 hours, you are a
freshman.
• If you have 30 or more hours, but less than
60 hours, you are a sophomore.
• If you have 60 or more hours, but less than
90 hours, you are a junior.
• If you have 90 or more hours, you are a
senior.
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C = f(H)
Numeric
Representation
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C = f(H)
Symbolic Representation
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C = f(H) Diagrammatic Representation
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C = f(H)
Graphical Representation
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Notes on Graphical Representation
• Vertical line test (p 39). To determine if
a graph represents a function, simply
visualize vertical lines in the xy-plane.
If each vertical line intersects a graph at
no more than one point, then it is the
output
graph of a function.
input
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1.4
Types of Functions
and Their Rates of
Change
♦ Identify and use constant and linear functions
♦ Interpret slope as a rate of change
♦ Identify and use nonlinear functions
♦ Recognize linear and nonlinear data
♦ Use and interpret average rate of change
♦ Calculate the difference quotient
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Constant Function
• A function f represented by f(x) = b,
where b is a constant (fixed number), is a
constant function.
Examples:
f(x) = 2
Note: Graph of a constant function is a horizontal line.
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Linear Function
• A function f represented by f(x) = ax + b,
where a and b are constants, is a linear function.
Examples:
f(x) = 2x + 3
Note that a f(x) = 2 is both a linear function and a constant function.
A constant function is a special case of a linear function.
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Rate of Change of a Linear Function
• Table of values for f(x) = 2x + 3.
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Slope of Line
• The slope m of the line passing through the
points (x1, y1) and (x2, y2) is
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Example of Calculation of Slope
• Find the slope of the line passing through the
points (2, 1) and (3, 9).
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Slide 1- 41
Example of a Nonlinear Function
• Table of values for
f(x) = x2
x y
0 0
1 1
2 4
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Average Rate of Change
• Let (x1, y1) and (x2, y2) be distinct points on the
graph of a function f. The average rate of
y  y1
change of f from x1 to x2 is 2
x2  x1
Note that the average rate of change of f from x1 to x2
is the slope of the line passing through
(x1, y1) and (x2, y2)
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The Difference Quotient
• The difference quotient of a function f is an
expression of the form f ( x  h)  f ( x)
h
where h is not 0.
Note that a difference
quotient is actually an
average rate of change.
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Example of Calculating a Difference Quotient
• Let f(x) = x2 + 3x. Find the difference
quotient of f and simplify the result.
f ( x  h)  f ( x) ( x  h) 2  3( x  h)  ( x 2  3x)


h
h
( x 2  2 xh  h 2 )  3x  3h  x 2  3x 2 xh  h 2  3h


h
h
h(2 x  h  3)
 2x  h  3
h
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