Unit: Functions and Mathematical Models Lesson: Introduction to
Math Analysis
Lessons 1.1-1.3 Introduction to Functions
Unit: Functions and Mathematical Models Lesson: Introduction to Functions
Functions: A function is a relation that pairs each element from the domain with exactly one element of the range
Algebraic: f(x)=
Graphic:
Numeric:
1
Math Analysis
Lessons 1.1-1.3 Introduction to Functions
Is it a function? Yes or No. If no, why not?
In function notation, the symbol f(x) is read f of x, x is called the independent variable. A value in the range of f is represented by the dependent variable y. Equation form:
Function Notation
2
Math Analysis
Lessons 1.1-1.3 Introduction to Functions
What does that 6 indicate?
Find
a)
b)
Domain: The domain of a function is the complete set of possible values of the independent variable The domain defines the horizontal boundary points of the function. Interval Notation:
3
Math Analysis
Lessons 1.1-1.3 Introduction to Functions
Domain:
Interval Notation:
The range defines the vertical boundary points of the function.
Range:
Interval Notation:
Finding Domains Algebraically
4
Math Analysis
Lessons 1.1-1.3 Introduction to Functions
5
Math Analysis
Lessons 1.1-1.3 Introduction to Functions
6
Math Analysis
Lessons 1.1-1.3 Introduction to Functions
Summary of Restricted Domains
fraction: radical: fraction with a radical in the denominator:
You Try -
1.
4.
2. 5. 3.
7
Math Analysis
Lessons 1.1-1.3 Introduction to Functions
Continuity: A graph is a continuous function if it has no breaks, holes or gaps. If it has one or more of those then it is discontinuous.
Types of Discontinuity A function has infinite discontinuity at x=c if the function value increases or decreases indefinitely as x approaches c from the left and right.
8
Math Analysis
Lessons 1.1-1.3 Introduction to Functions
Types of Discontinuity A function has jump discontinuity at x=c if the limit of the function as x approaches c from the left and right exist but have two distinct value.
Types of Discontinuity A function has a point discontinuity if the function is continuous everywhere except for the hole at x=c
9
Math Analysis
Lessons 1.1-1.3 Introduction to Functions
State if the function is continuous or discontinuous. If it is discontinuous state c.
State if the function is continuous or discontinuous. If it is discontinuous state c.
10
Math Analysis
Lessons 1.1-1.3 Introduction to Functions
End Behavior: the end behavior of a function describes how a function behaves at either ends of the graph. Left End Behavior what is happening at the extreme left of the graph.
Calculus Notation: Pre Calculus Notation:
As
Right End Behavior what is happening at the extreme right of the graph.
Calculus Notation: Pre Calculus Notation:
As
End Behavior
11
Math Analysis
Lessons 1.1-1.3 Introduction to Functions
End Behavior
End Behavior
12
Math Analysis
Lessons 1.1-1.3 Introduction to Functions
Even Function: A function f is even if the graph of f is symmetric with respect to the yaxis. Algebraically, f is even if and only if f(-x) = f(x) for all x in the domain of f.
Odd Function: A function f is odd if the graph of f is symmetric with respect to the origin. Algebraically, f is odd if and only if f(-x) = -f(x) for all x in the domain of f.
13
Math Analysis
Lessons 1.1-1.3 Introduction to Functions
Is the function odd even or neither?
State the Type of Symmetry
14
Math Analysis
Lessons 1.1-1.3 Introduction to Functions
Unit: Functions and Mathematical Models Lesson: Introduction to Functions Day 2
Increasing and Decreasing
Increasing: A function is in increasing on an interval if and only if for any two points in the interval, a positive change in x results in a positive change in f(x).
15
Math Analysis
Lessons 1.1-1.3 Introduction to Functions
Decreasing: A function is in decreasing on an interval if and only if for any two points in the interval, a positive change in x results in a negative change in f(x).
The challenge comes in when the function is both increasing and decreasing.
16
Math Analysis
Lessons 1.1-1.3 Introduction to Functions
State the increasing and decreasing intervals
State the increasing and decreasing intervals
17
Math Analysis
Lessons 1.1-1.3 Introduction to Functions
State the increasing and decreasing intervals
(-2,2) (0.5,-1.25)
(-1,7)
The point we labeled on the graph are the extrema. They are the points at which the function changes its increasing or decreasing behavior. The points themselves are called critical points. Any maximum or minimums MUST occur at a critical point.
18
Math Analysis
Lessons 1.1-1.3 Introduction to Functions
Relative and Absolute Maximum: A relative maximum of a function f is the greatest value f(x) can have on some interval of the domain. Quartic - negative leading coefficient
If a relative maximum is the greatest value the function can attain over its entire domain, then it is the absolute maximum.
Relative and Absolute Minimum: A relative minimum of a function f is the least value f(x) can have on some interval of the domain. If a relative minimum is the least value the function can attain over its entire domain, then it is the absolute minimum. Quartic - positive leading coefficient
19
Math Analysis
Lessons 1.1-1.3 Introduction to Functions
State all of the extrema
20
Math Analysis
Lessons 1.1-1.3 Introduction to Functions
State all of the extrema and the increasing or decreasing intervals.
What is the slope?
21
Math Analysis
Lessons 1.1-1.3 Introduction to Functions
To find the slope on an interval of a curve you need to use one of two formulas:
Average Rate of Change =
or
Find the Average Rate of Change
on the interval
22
Math Analysis
Lessons 1.1-1.3 Introduction to Functions
Find the Average Rate of Change
on the interval
Given
find the average rate of change formula
23
Math Analysis
Lessons 1.1-1.3 Introduction to Functions
Given
find the average rate of change formula
Find the average rate of change for the function
24
Math Analysis
Lessons 1.1-1.3 Introduction to Functions
Evaluate that rate of change on the interval [2, 2.01] and [-3, -3.01]
Unit: Functions and Mathematical Models Lesson: Introduction to Functions Day 3
25
Math Analysis
Lessons 1.1-1.3 Introduction to Functions
For all of our problems today we will find the equation for the average rate of change and then evaluate it on a given interval.
on the interval [1, 1.1]
on the interval [0.5, 0.51]
26
Math Analysis
Lessons 1.1-1.3 Introduction to Functions
on the interval [1, 1.1]
on the interval [1.9, 2]
27
Math Analysis
Lessons 1.1-1.3 Introduction to Functions
on the interval [3,3.1]
28
Lessons 1.1-1.3 Introduction to Functions
Unit: Functions and Mathematical Models Lesson: Introduction to Functions
Functions: A function is a relation that pairs each element from the domain with exactly one element of the range
Algebraic: f(x)=
Graphic:
Numeric:
1
Math Analysis
Lessons 1.1-1.3 Introduction to Functions
Is it a function? Yes or No. If no, why not?
In function notation, the symbol f(x) is read f of x, x is called the independent variable. A value in the range of f is represented by the dependent variable y. Equation form:
Function Notation
2
Math Analysis
Lessons 1.1-1.3 Introduction to Functions
What does that 6 indicate?
Find
a)
b)
Domain: The domain of a function is the complete set of possible values of the independent variable The domain defines the horizontal boundary points of the function. Interval Notation:
3
Math Analysis
Lessons 1.1-1.3 Introduction to Functions
Domain:
Interval Notation:
The range defines the vertical boundary points of the function.
Range:
Interval Notation:
Finding Domains Algebraically
4
Math Analysis
Lessons 1.1-1.3 Introduction to Functions
5
Math Analysis
Lessons 1.1-1.3 Introduction to Functions
6
Math Analysis
Lessons 1.1-1.3 Introduction to Functions
Summary of Restricted Domains
fraction: radical: fraction with a radical in the denominator:
You Try -
1.
4.
2. 5. 3.
7
Math Analysis
Lessons 1.1-1.3 Introduction to Functions
Continuity: A graph is a continuous function if it has no breaks, holes or gaps. If it has one or more of those then it is discontinuous.
Types of Discontinuity A function has infinite discontinuity at x=c if the function value increases or decreases indefinitely as x approaches c from the left and right.
8
Math Analysis
Lessons 1.1-1.3 Introduction to Functions
Types of Discontinuity A function has jump discontinuity at x=c if the limit of the function as x approaches c from the left and right exist but have two distinct value.
Types of Discontinuity A function has a point discontinuity if the function is continuous everywhere except for the hole at x=c
9
Math Analysis
Lessons 1.1-1.3 Introduction to Functions
State if the function is continuous or discontinuous. If it is discontinuous state c.
State if the function is continuous or discontinuous. If it is discontinuous state c.
10
Math Analysis
Lessons 1.1-1.3 Introduction to Functions
End Behavior: the end behavior of a function describes how a function behaves at either ends of the graph. Left End Behavior what is happening at the extreme left of the graph.
Calculus Notation: Pre Calculus Notation:
As
Right End Behavior what is happening at the extreme right of the graph.
Calculus Notation: Pre Calculus Notation:
As
End Behavior
11
Math Analysis
Lessons 1.1-1.3 Introduction to Functions
End Behavior
End Behavior
12
Math Analysis
Lessons 1.1-1.3 Introduction to Functions
Even Function: A function f is even if the graph of f is symmetric with respect to the yaxis. Algebraically, f is even if and only if f(-x) = f(x) for all x in the domain of f.
Odd Function: A function f is odd if the graph of f is symmetric with respect to the origin. Algebraically, f is odd if and only if f(-x) = -f(x) for all x in the domain of f.
13
Math Analysis
Lessons 1.1-1.3 Introduction to Functions
Is the function odd even or neither?
State the Type of Symmetry
14
Math Analysis
Lessons 1.1-1.3 Introduction to Functions
Unit: Functions and Mathematical Models Lesson: Introduction to Functions Day 2
Increasing and Decreasing
Increasing: A function is in increasing on an interval if and only if for any two points in the interval, a positive change in x results in a positive change in f(x).
15
Math Analysis
Lessons 1.1-1.3 Introduction to Functions
Decreasing: A function is in decreasing on an interval if and only if for any two points in the interval, a positive change in x results in a negative change in f(x).
The challenge comes in when the function is both increasing and decreasing.
16
Math Analysis
Lessons 1.1-1.3 Introduction to Functions
State the increasing and decreasing intervals
State the increasing and decreasing intervals
17
Math Analysis
Lessons 1.1-1.3 Introduction to Functions
State the increasing and decreasing intervals
(-2,2) (0.5,-1.25)
(-1,7)
The point we labeled on the graph are the extrema. They are the points at which the function changes its increasing or decreasing behavior. The points themselves are called critical points. Any maximum or minimums MUST occur at a critical point.
18
Math Analysis
Lessons 1.1-1.3 Introduction to Functions
Relative and Absolute Maximum: A relative maximum of a function f is the greatest value f(x) can have on some interval of the domain. Quartic - negative leading coefficient
If a relative maximum is the greatest value the function can attain over its entire domain, then it is the absolute maximum.
Relative and Absolute Minimum: A relative minimum of a function f is the least value f(x) can have on some interval of the domain. If a relative minimum is the least value the function can attain over its entire domain, then it is the absolute minimum. Quartic - positive leading coefficient
19
Math Analysis
Lessons 1.1-1.3 Introduction to Functions
State all of the extrema
20
Math Analysis
Lessons 1.1-1.3 Introduction to Functions
State all of the extrema and the increasing or decreasing intervals.
What is the slope?
21
Math Analysis
Lessons 1.1-1.3 Introduction to Functions
To find the slope on an interval of a curve you need to use one of two formulas:
Average Rate of Change =
or
Find the Average Rate of Change
on the interval
22
Math Analysis
Lessons 1.1-1.3 Introduction to Functions
Find the Average Rate of Change
on the interval
Given
find the average rate of change formula
23
Math Analysis
Lessons 1.1-1.3 Introduction to Functions
Given
find the average rate of change formula
Find the average rate of change for the function
24
Math Analysis
Lessons 1.1-1.3 Introduction to Functions
Evaluate that rate of change on the interval [2, 2.01] and [-3, -3.01]
Unit: Functions and Mathematical Models Lesson: Introduction to Functions Day 3
25
Math Analysis
Lessons 1.1-1.3 Introduction to Functions
For all of our problems today we will find the equation for the average rate of change and then evaluate it on a given interval.
on the interval [1, 1.1]
on the interval [0.5, 0.51]
26
Math Analysis
Lessons 1.1-1.3 Introduction to Functions
on the interval [1, 1.1]
on the interval [1.9, 2]
27
Math Analysis
Lessons 1.1-1.3 Introduction to Functions
on the interval [3,3.1]
28
