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Learn the Important Underlying Concept of Functions in Mathematics

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- 22.5 hours on-demand video
- 1 Article
- Full lifetime access
- Access on mobile and TV

- Certificate of Completion

What Will I Learn?

- Have a strong foundation and understanding of functions
- Have a strong foundation of the underlying concepts of calculus

Requirements

- High School Basic Algebra

Description

This course teaches you all the important underlying concepts in functions in Mathematics. The knowledge that you gain here can be further completed in our next courses towards a complete mastery of calculus.

This course covers the following topics:

- Function and Function Notation
- Domain and Range of Functions
- Rates of Change and Behavior of Graphs
- Composition of Functions
- Transformation of Functions
- Absolute Value Functions
- Inverse Functions

As described above, this course can also be taken in combination with our other courses in this course series. If you're interested in learning mathematics with us all the way up to calculus, please read our "Mathematics" page on "Greatitcourses" website.

Who is the target audience?

- This course is for you if you're planning a career in Engineering, Programming, Science, etc. where you'd need a strong foundation in mathematics.
- This course is also for you if you're changing careers and need to use the discussed concepts in your future career, for example as a computer programmer.

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Curriculum For This Course

130 Lectures

22:33:02
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3.0 - Introduction
6 Lectures
23:01

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3.1 - Functions and Function Notation
18 Lectures
03:56:53

Here we’ll have a short discussion about the topics covered in this course, how to get a copy of the textbook if you had any problems finding that and the section in the book we’re covering in this course. We’ll also show you where this course is on the main road map on greatitcourses.com. |

Preview
06:29

Here, we’ll have a discussion and a general overview of the concept of functions to build a foundation upon which we can build the rest of the course.

Preview
17:40

Preview
19:34

Here, we’ll have a typical coffee shop menu and based on that decide whether price is a function of item and vice versa.

Preview
10:54

Here, we’ll solve two examples, first a typical average grade system and second a list of famous baseball players of all time. In each case, we decide whether one set can be considered a function of the other one and vice versa.

How to Determine If a Relation a Function - Exercise

10:18

In this video, we’ll talk about function notation like f(x). We learn how and in what situations we can use that and what is basically represented by it. Moreover, we’ll do some examples to learn the concept better.

Function Notation

19:41

In this video, we’ll study some relationships between pairs of numbers represented in form of tables. Each table has a set of inputs and outputs. After studying each table, we decide whether the table represents a function or not.

Representing Functions Using Tables

10:00

In this video, you’ll learn how to evaluate functions at different points for inputs. We’ll also learn what the inverse of this situation would look like. Moreover, we’ll go through some example to solidify the concepts learned here.

Finding Input and Output Values of a Function

15:58

In this video, you’ll learn how to figure out the input values in a function that created a specific output.

Evaluating Functions - Exercise

12:29

In this video, you’ll learn how to determine whether a variable in a formula is a function of the other variable. For example, in the formula x² + y² = 1, you can determine whether y is a function of x or not.

Evaluating a Function Expressed in Formulas

15:38

In this video, you’ll learn how to evaluate function values by reading the information related to the function represented in a tabular form. Each input in the table corresponds to an output and so based on that, you can evaluate any desired input.

Evaluating a Function Given in Tabular Form

09:15

In this video, you’ll learn how to use the graph of a function in order to determine the output value related to any input value present in the domain of the function. For example, by looking at the graph of a function, you can identify which output is related to the input x =4 and based on that you can write, f(4) = “that identified output value read on the graph”. You can also do this to solve equations like f(x) = 2, in which you could identify for which value/values of x, you would have an output value of 2.

Finding Function Values from a Graph

08:23

In this video, you’ll learn how to determine whether a function is one-to-one, meaning that each output value corresponds exactly to one input value and that there are no repeated x or y values. We’ll also do some exercises to understand the concept better.

Determining Whether a Function is One-to-One

17:49

In this video, you’ll learn how you can use a vertical line, run it through the domain of a function parallel to the y axis to decide whether a graph represent a function or not.

Vertical Line Test

11:12

In this video, you’ll learn how you can use a horizontal line, run it through the range of a function, parallel to the x axis, to decide whether a function is one-to-one or not.

Horizontal Line Test

14:29

In this video, we’ll graph some toolkit functions like f(x) = x². These are the kind of functions that are used very often in mathematics. We’ll also talk about some important characteristics of them, like vertex, minimum, maximum, domain and range.

Basic Toolkit Functions (Part 1 of 3)

17:36

In this video, we’ll graph some toolkit functions like f(x) = x². These are the kind of functions that are used very often in mathematics. We’ll also talk about some important characteristics of them, like vertex, minimum, maximum, domain and range.

Basic Toolkit Functions (Part 2 of 3)

14:00

In this video, we’ll graph some toolkit functions like f(x) = x². These are the kind of functions that are used very often in mathematics. We’ll also talk about some important characteristics of them, like vertex, minimum, maximum, domain and range.

Basic Toolkit Functions (Part 3 of 3)

05:28

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3.2 - Domain and Range
12 Lectures
02:42:11

Domain and Range Introduction

12:23

In this video, we’ll give you a summery of the interval notation. For example, x > a on a number line in interval notation as (a, infinity). We’ll also do a few examples finding the domains of a few functions.

Interval Notation Summery

15:30

In this video, you’ll learn how to find the domain of a function. We’ll also do a few exercises to learn the concept properly.

Finding the Domain of a Function - Exercise

15:26

In this video, you’ll learn how to find the domain of a function. We’ll also do a few exercises to learn the concept properly.

Finding the Domain of a Function - Exercise

16:14

In this video, you’ll learn how to use inequality notation, set-builder notation and the interval notation to specify domain and range of a function.

Inequality, Set-builder and Interval Notations

13:03

In this video, you’ll learn how to describe sets on a real-number line in set-builder, inequality and interval notations.

Describing Set on the Real-Number Line

08:37

In this video, you’ll learn how to find the domain and range of functions based on the graph of the function. We’ll also do some examples to learn the concept better.

Finding Domain and Range From Graphs

19:01

In this video, we’ll study the domains and ranges of some famous toolkit functions.

Finding Domains and Ranges of the Toolkit Functions

19:17

In this video, we’ll find the domain and range of square root functions.

Finding the Domain and Range of Functions - Exercise

06:43

In this video, you’ll learn what piecewise-defined functions are and in what sort of situations they can be used.

Piecewise-Defined Functions

16:06

p190, e12 - In this video, you’ll learn how to interpret and graph a piecewise function that represents how a cell phone company charges their customers for data transfer.

Working with a Piecewise Function - Exercise

10:34

p191, e13 - In this video, we’ll draw the graph of a piecewise functions that consists of a three pieces.

Graphing a Piecewise Function - Exercise

09:17

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3.3 - Rates of Change and Behavior of Graphs
14 Lectures
02:12:16

In this video, we’ll introduce the concept of rate of change. We’ll use an example of the average cost of a gallon of gasoline in dollars over the period of 7 years from 2005 to 2012 and study the change in cost over different periods between the two years.

Rate of Change Introduction

13:11

In this video, we'll learn the practical meaning of average rate of change using a practical example.

Rate of Change Formula and Example

09:03

Computing an Average Rate of Change based on a Table

14:15

Computing an Average Rate of Change based on a Function Graph

12:26

Computing an Average Rate of Change based on a Function Formula

10:48

In this video, we'll go through two interesting examples of calculating the average rate of change.

Computing an Average Rate of Change - Example

12:40

Increasing, Decreasing, Local Minimum, Maximum, Extrema - Part 1

14:57

Increasing, Decreasing, Local Minimum, Maximum, Extrema - Part 2

03:56

In this video, we're going to analyze the graph of a function and identify in what intervals the function is increasing or decreasing.

Finding Increasing and Decreasing Intervals on a Graph

06:23

Finding Extrema of a Function using Technology

11:36

Finding the Local Maximum and Local Minimum of a Function - Example

01:56

Analyzing the Toolkit Functions for Increasing and Decreasing Intervals

09:34

Absolute Minimum and Absolute Maximum of a Function

06:37

Finding the Absolute Minima and Maxima of a Function - Example

04:54

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3.4 - Composition of Functions
14 Lectures
02:07:57

Composition of Functions - Concept

14:24

Combining Functions Using Algebraic Operations

12:55

Combining Functions Using Algebraic Operations - Example

07:40

Composition of Functions - Concept and Mathematical Notation

12:36

Composition of Functions Not Commutative - Exercise

04:39

Composition of Functions - Exercise

04:16

Composition of Functions - Exercise

07:06

Evaluating Composite Functions Using Tables

05:59

Evaluating Composite Functions Using Graphs

06:31

Evaluating Composite Functions Using Function Formulas

10:07

Finding the Domain of a Composite Function - Part 1

14:12

Finding the Domain of a Composite Function - Part 2

09:49

Finding the Domain of a Composite Function - Exercise

08:20

Decomposing Composite Functions

09:23

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3.5 - Transformation of Functions
40 Lectures
06:52:24

Transformation of Functions - Introduction

13:08

Vertical Transformation of Functions

11:51

Vertical Shift - Example

05:24

Vertical Shift - Example

07:19

Horizontal Shift of Functions

14:19

Horizontal Shift to Right and Left - Part 1

14:30

Horizontal Shift to Right and Left - Part 2

05:30

Horizontal Shift - Example

07:53

Inside and Outside Changes Affecting Domain or Range

06:34

Shifting a Tabular Function Horizontally

05:57

Identifying a Horizontal Shift of a Toolkit Function

08:06

Horizontal Function shift - Example

06:40

Graphing Combined Vertical and Horizontal Shifts - Example

11:03

Identifying Combined Horizontal and Vertical Shifts - Example

10:44

In this video, we're moving the toolkit reciprocal function to the right and up. We'll then find the formula of the transformed function, draw the graph of both functions and compare them.

Transformation of Functions - Example

06:46

Reflecting Function About the x and y axes - Introduction

11:07

Reflecting a Graph Horizontally and Vertically - Example

10:43

Reflecting a Graph Horizontally and Vertically - Example

10:58

Reflecting a Tabular Function Horizontally and Vertically - Example

04:59

In this video, we're developing a function that represents a learning model. The function is a transformation of one of the toolkit functions. Three steps of transformation are applied to the function.

Applying a Learning Model Equation - Example

13:11

In this video, we're going to graph the vertical and horizontal transformations of the toolkit function f(x) = x^2

Graphing the Vertical and Horizontal Transformations of a Toolkit Function

06:44

Even and Odd Functions - Part 1

14:46

Even and Odd Functions - Part 2

08:26

In this video, we have a function in the form of a polynomial of degree 4. We'll verify whether the polynomial is even or odd.

Even and Odd Functions - Example

08:38

Vertical and Horizontal Compression and Stretches - Part 1

09:59

Vertical and Horizontal Compression and Stretches - Part 2

13:38

Vertical and Horizontal Compression and Stretches - Part 3

13:27

p234

Vertical Stretch of Functions and Example

12:37

p236

Finding a Vertical Compression of a Tabular Function

07:33

p236

Recognizing a Vertical Stretch - Example

07:52

p237 - In this video, we're taking the identity toolkit function, f(x) = x and stretch the function and move it down as well. We'll create a formula for the new function and draw the graph as well.

Function Formula Transformation - Example

07:30

p237

Horizontal Compression and Stretches - Concept

14:47

p238 - In this video, we'll create a new function based on a population of fruit flies. The new function created based on the original function progresses through its life span twice as fast as the original function and so it represents a horizontal compression.

Graphing a Horizontal Compression - Example

14:41

p238 - In this video, we'll create a new function based on a population of fruit flies. The new function created based on the original function progresses through its life span twice as fast as the original function and so it represents a horizontal compression. This is the same problem as the last video. It has been solved in a different way.

Graphing a Horizontal Compression - Example

14:06

p238, e17

Finding a Horizontal Stretch for a Tabular Function

12:44

p239, e18

Recognizing a Horizontal Compression on a Graph

06:19

p239, ti11 - In this video, we take the toolkit square root function and stretch it by a factor of 3. We then find the formula of the stretched function.

Finding the Formula for a Stretched Toolkit Function

14:00

p240

Performing a Sequence of Transformations

13:29

p240, e19

Finding a Triple Transformation of a Tabular Function - Example

09:30

p241, e20

Finding a Triple Transformation of a Graph

14:56

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3.6 - Absolute Value Functions
7 Lectures
01:07:10

p247

Absolute Value Function - Introduction

14:20

p247

Absolute Value - Example

03:54

p248

Graphing an Absolute Value Function

09:42

p248, e2

Writing an Equation for an Absolute Value Function Given a Graph

11:44

p249, ti2

Absolute Value Function Transformation - Example

07:06

p251, e3

Finding the Zeros of an Absolute Value Function - Example

12:25

p251, ti3

Solving an Absolute Value Equation

07:59

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3.7 - Inverse Functions
19 Lectures
03:11:10

p254

Inverse Functions - Introduction

13:59

Domain, Codomain, Range and Image of a Function

11:36

One-to-one or Injective Functions

05:35

Onto or Surjective Functions

08:15

Bijective or One-to-one Correspondence Functions

13:16

p256, e1

Identifying an Inverse Function for a Given Input-Output Pair - Example

10:00

p256, ti1

Original and Inverse Functions Input-Output Pairs

01:39

p256, e2

Testing Inverse Relationships Algebraically - Example

07:12

p257

Finding the Domain and Range of Inverse Functions

13:31

p258, e4

Finding the Inverse of Toolkit Functions - Part 1

13:49

p258, e4

Finding the Inverse of Toolkit Functions - Part 2

11:08

p259

Inverting Tabular Functions

05:25

p260, e6

Evaluating a Function and its Inverse from a Graph at Specific Points

07:54

p260

Finding Inverses of Functions Represented by Formulas - Part 1

13:14

Finding Inverses of Functions Represented by Formulas - Part 2

13:51

p260, e7

Inverting the Fahrenheit-to-Celsius Function - Example

07:40

p261, e8

Solving to Find an Inverse Function

11:31

p261, e9

Solving to Find an Inverse with Radicals

11:49

p262, e10

Finding the Inverse of a Function Using Reflection about the Identity Line

09:46

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