Your cart is empty.
Keep shopping
Teach on Udemy
Turn what you know into an opportunity and reach millions around the world.
Learn MoreAlgebra and Trigonometry: Functions
Rating: 4.5 out of 54.5(15 ratings)
840 students
Algebra and Trigonometry: Functions
Learn the Important Underlying Concept of Functions in Mathematics
Created byGreat IT Courses
Last updated 6/2017
English
English [Auto],
What you'll learn
- Have a strong foundation and understanding of functions
- Have a strong foundation of the underlying concepts of calculus
Explore related topics
Course content
8 sections • 130 lectures • 22h 33m total length
- Instructor Introduction2:28
- Course Curriculum4:17
The algebra and trigonometry curriculum uses a 13-chapter textbook, with each chapter forming one course; find your chapter on the course page and download the book.
- Road Map6:20
Build a strong foundation from algebra and trigonometry through calculus i–iii with the math road map. Connect concepts to real life and plan your study within the core math series.
- How to Best Use This Course5:14
Learn with two video types: concept videos and exercise videos, master each concept, complete end-of-section exercises, and engage with the discussion board to progress through six sections toward calculus mastery.
- Feedback4:36
Explore the course feedback system, rate videos with stars, and share your view on the algebra and trigonometry content, while noting the instructor’s commitment to correctness and corrections.
- Legal Disclaimer0:06
- Course Introduction6:29
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. - Introduction to Functions (Part 1 of 2)17:40
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.
- Introduction to Functions (Part 2 of 2)19:34
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.
- How to Determine If a Relation a Function - Exercise10:54
Here, we’ll have a typical coffee shop menu and based on that decide whether price is a function of item and vice versa.
- How to Determine If a Relation a Function - Exercise10:18
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.
- Function Notation19:41
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.
- Representing Functions Using Tables10:00
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.
- Finding Input and Output Values of a Function15:58
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.
- Evaluating Functions - Exercise12:29
In this video, you’ll learn how to figure out the input values in a function that created a specific output.
- Evaluating a Function Expressed in Formulas15:38
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 Given in Tabular Form9:15
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.
- Finding Function Values from a Graph8:23
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.
- Determining Whether a Function is One-to-One17:49
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.
- Vertical Line Test11:12
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.
- Horizontal Line Test14:29
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.
- 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)5:28
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.
- Domain and Range Introduction12:23
Explore how to determine the domain and range of functions defined by equations and piecewise definitions, using interval notation with inclusive and exclusive bounds.
- Interval Notation Summery15:30
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.
- Finding the Domain of a Function - Exercise15:26
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 - Exercise16:14
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.
- Inequality, Set-builder and Interval Notations13:03
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.
- Describing Set on the Real-Number Line8:37
In this video, you’ll learn how to describe sets on a real-number line in set-builder, inequality and interval notations.
- Finding Domain and Range From Graphs19:01
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 Domains and Ranges of the Toolkit Functions19:17
In this video, we’ll study the domains and ranges of some famous toolkit functions.
- Finding the Domain and Range of Functions - Exercise6:43
In this video, we’ll find the domain and range of square root functions.
- Piecewise-Defined Functions16:06
In this video, you’ll learn what piecewise-defined functions are and in what sort of situations they can be used.
- Working with a Piecewise Function - Exercise10:34
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.
- Graphing a Piecewise Function - Exercise9:17
p191, e13 - In this video, we’ll draw the graph of a piecewise functions that consists of a three pieces.
- Rate of Change Introduction13:11
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 Formula and Example9:03
In this video, we'll learn the practical meaning of average rate of change using a practical example.
- Computing an Average Rate of Change based on a Table14:15
Compute the average rate of change from a table, using 2007 and 2009 gasoline prices, and interpret the negative slope as a price decrease in dollars per year.
- Computing an Average Rate of Change based on a Function Graph12:26
Calculate the average rate of change from a function graph by comparing f(-1)=4 and f(2)=1, yielding a slope of -1 for the interval [-1, 2].
- Computing an Average Rate of Change based on a Function Formula10:48
Explore computing the average rate of change for a function from its formula, using delta y over delta x, and evaluate f at chosen points with domain notes.
- Computing an Average Rate of Change - Example12:40
In this video, we'll go through two interesting examples of calculating the average rate of change.
- Increasing, Decreasing, Local Minimum, Maximum, Extrema - Part 114:57
Learn to read graphs to identify where functions increase, decrease, and attain local maxima or minima, and understand extrema and constant regions with simple examples.
- Increasing, Decreasing, Local Minimum, Maximum, Extrema - Part 23:56
Analyze local minimum and local maximum in functions, understand extrema on intervals, and connect increasing and decreasing behavior to calculus applications.
- Finding Increasing and Decreasing Intervals on a Graph6:23
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 Extrema of a Function using Technology11:36
Find local extrema of a rational function with a vertical asymptote at zero by using technology to plot and compute precise maximum and minimum points, noting the domain excludes zero.
- Finding the Local Maximum and Local Minimum of a Function - Example1:56
- Analyzing the Toolkit Functions for Increasing and Decreasing Intervals9:34
Analyze toolkit functions' increasing and decreasing intervals, including constant, identity, quadratic, cubic, reciprocal, reciprocal square, cube root, square root, and absolute value, and discuss domains and absolute maximum and minimum.
- Absolute Minimum and Absolute Maximum of a Function6:37
Identify the absolute extrema: the maximum occurs when f(c) ≥ f(x) for all x in the domain, and the minimum when f(d) ≤ f(x) for all x in the domain.
- Finding the Absolute Minima and Maxima of a Function - Example4:54
Identify absolute maxima and minima from a graph using the sine function, noting y = 1 and y = -1 at key x-values across the domain.
- Composition of Functions - Concept14:24
learn to compose functions by merging smaller functions into a new rule and evaluate the resulting composite function. see how to decompose composite functions into components and determine their domain.
- Combining Functions Using Algebraic Operations12:55
Combine functions using addition, subtraction, multiplication, and division with matching input and output units. Understand how to pair year-based functions to compute totals, such as household income, while preserving units.
- Combining Functions Using Algebraic Operations - Example7:40
Learn how to combine functions with algebraic operations and compute compositions like f(g(x)) and g(f(x)). Simplify expressions, handle domain restrictions (where denominators can't be zero), and compare resulting function degrees.
- Composition of Functions - Concept and Mathematical Notation12:36
Explore composition of functions by applying the inner function first, then the outer, noting domain and range, and how f∘g differs from g∘f.
- Composition of Functions Not Commutative - Exercise4:39
Explore when the composition of functions is commutative by evaluating examples f(x)=2x+1 and g(x)=-x; compare f∘g and g∘f to determine noncommutativity.
- Composition of Functions - Exercise4:16
Explore the composition of functions by interpreting C(S(t)) as calories burned from sit-ups completed in a given number of minutes, and ensure the inputs and results make sense.
- Composition of Functions - Exercise7:06
- Evaluating Composite Functions Using Tables5:59
Learn to evaluate composite functions using tables by identifying inner outputs and substituting them into outer functions, then work from inside out with examples of f and g.
- Evaluating Composite Functions Using Graphs6:31
- Evaluating Composite Functions Using Function Formulas10:07
- Finding the Domain of a Composite Function - Part 114:12
Determine the domain of a composite function by ensuring the input is in the domain of g and the output lies in the domain of f.
- Finding the Domain of a Composite Function - Part 29:49
Determine the domain of a composite function by excluding x values that make the outer function undefined or the inner input invalid, such as two thirds and two.
- Finding the Domain of a Composite Function - Exercise8:20
Find the domain of the composite function f(g(x)) where f(x)=1/(x-2) and g(x)=√(x+4); the radical requires x≥−4 and x≠0 to avoid division by zero, yielding [-4,0) ∪ (0,∞).
- Decomposing Composite Functions9:23
- Transformation of Functions - Introduction13:08
- Vertical Transformation of Functions11:51
- Vertical Shift - Example5:24
learn the vertical shift of a function, where the output is moved down by a constant while the input remains unchanged, illustrated with f(x) and g(x).
- Vertical Shift - Example7:19
The example shows a vertical shift by lifting the height function by 10 meters, defining b(t) = h(t) + 10 where h(t) = -4.9 t^2 + 30 t.
- Horizontal Shift of Functions14:19
Explore how horizontal shifts move graphs using f(x) and f(x-h) to the right for positive h and to the left for negative h.
- Horizontal Shift to Right and Left - Part 114:30
Explore how horizontal shifts move a function left or right using f(x)=x and its x+2 and x−2 variants, showing how input changes preserve outputs.
- Horizontal Shift to Right and Left - Part 25:30
- Horizontal Shift - Example7:53
Explore horizontal shifts of a function, showing left shifts via f(t+2) and right shifts via f(t-2), and how input domains change while outputs remain the same.
- Inside and Outside Changes Affecting Domain or Range6:34
Learn how inside changes to a function shift the input and the domain, while outside changes shift the output and the range, with examples of horizontal and vertical translations.
- Shifting a Tabular Function Horizontally5:57
Shift a tabular function horizontally to the right by 3 units, creating a new function with inputs 3 units larger and the same outputs as the original.
- Identifying a Horizontal Shift of a Toolkit Function8:06
Identify a horizontal shift of the toolkit function f(x)=x^2 by moving the vertex from (0,0) to (2,0), yielding g(x)=(x-2)^2, without distortion.
- Horizontal Function shift - Example6:40
Explore the horizontal shift of f(x)=sqrt(x) to g(x)=f(x+2). See left shift by two units and compare original and transformed graphs.
- Graphing Combined Vertical and Horizontal Shifts - Example11:03
Combine horizontal and vertical shifts to transform a function, distinguishing inside (domain) from outside (range) changes, and illustrate with the absolute value function yielding y = |x+1| - 3.
- Identifying Combined Horizontal and Vertical Shifts - Example10:44
Identify combined horizontal and vertical shifts of the square root function. Transform f(x)=√x to h(x)=√(x−1)+2 by shifting right 1 and up 2, changing domain to [1,∞) and range to [2,∞).
- Transformation of Functions - Example6:46
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.
- Reflecting Function About the x and y axes - Introduction11:07
Explore vertical and horizontal reflections of a base function about the x-axis and y-axis; learn how g(x)=f(-x) describes horizontal reflection, and how vertical reflection preserves x-values while negating y.
- Reflecting a Graph Horizontally and Vertically - Example10:43
Explore reflecting the square root function vertically and horizontally, deriving p(t) as minus sqrt(t) and sqrt(-t), and note how the range and domain change.
- Reflecting a Graph Horizontally and Vertically - Example10:58
Explore how reflecting and translating the absolute value function |x-1| yields horizontal shifts and vertical reflections, illustrated by g(x) = -|x-1| and the horizontal reflection f(-x).
- Reflecting a Tabular Function Horizontally and Vertically - Example4:59
Explore how to reflect a tabular function vertically and horizontally, using g(x) = -f(x) for vertical reflections and x-values negated for horizontal reflections, with example inputs and outputs.
- Applying a Learning Model Equation - Example13:11
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.
- Graphing the Vertical and Horizontal Transformations of a Toolkit Function6:44
In this video, we're going to graph the vertical and horizontal transformations of the toolkit function f(x) = x^2
- Even and Odd Functions - Part 114:46
- Even and Odd Functions - Part 28:26
Classify functions as even or odd by testing f(-x) with f(x) and -f(x); recognize even symmetry about the y-axis and odd symmetry about the origin, with f(x)=x^3+2x as an example.
- Even and Odd Functions - Example8:38
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.
- Vertical and Horizontal Compression and Stretches - Part 19:59
This lecture introduces vertical and horizontal compression and stretches, showing how outside changes shift graphs up, down, left, or right while preserving shape.
- Vertical and Horizontal Compression and Stretches - Part 213:38
Explore vertical scaling of functions with outside changes like 0.5 f(x) and 2 f(x), reflect about the x-axis with negative factors; examine inside changes such as f(0.5x) and f(-2x).
- Vertical and Horizontal Compression and Stretches - Part 313:27
Explore how horizontal compression and vertical stretches transform functions, showing how inside versus outside changes alter input/output, domain, and graph shape.
- Vertical Stretch of Functions and Example12:37
p234
- Finding a Vertical Compression of a Tabular Function7:33
p236
- Recognizing a Vertical Stretch - Example7:52
p236
- Function Formula Transformation - Example7:30
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.
- Horizontal Compression and Stretches - Concept14:47
p237
- Graphing a Horizontal Compression - Example14:41
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.
- Graphing a Horizontal Compression - Example14:06
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.
- Finding a Horizontal Stretch for a Tabular Function12:44
p238, e17
- Recognizing a Horizontal Compression on a Graph6:19
p239, e18
- Finding the Formula for a Stretched Toolkit Function14:00
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.
- Performing a Sequence of Transformations13:29
p240
- Finding a Triple Transformation of a Tabular Function - Example9:30
p240, e19
- Finding a Triple Transformation of a Graph14:56
p241, e20
- Absolute Value Function - Introduction14:20
p247
- Absolute Value - Example3:54
p247
- Graphing an Absolute Value Function9:42
p248
- Writing an Equation for an Absolute Value Function Given a Graph11:44
p248, e2
- Absolute Value Function Transformation - Example7:06
p249, ti2
- Finding the Zeros of an Absolute Value Function - Example12:25
p251, e3
- Solving an Absolute Value Equation7:59
p251, ti3
- Inverse Functions - Introduction13:59
p254
- Domain, Codomain, Range and Image of a Function11:36
Explore the concepts of a function's domain, codomain, and range (image), and see how inputs map to outputs through simple examples.
- One-to-one or Injective Functions5:35
- Onto or Surjective Functions8:15
Explore surjective (onto) functions, where every codomain element is mapped by at least one domain element, making the range equal to the codomain and distinguishing them from non-surjective cases.
- Bijective or One-to-one Correspondence Functions13:16
- Identifying an Inverse Function for a Given Input-Output Pair - Example10:00
p256, e1
- Original and Inverse Functions Input-Output Pairs1:39
p256, ti1
- Testing Inverse Relationships Algebraically - Example7:12
p256, e2
- Finding the Domain and Range of Inverse Functions13:31
p257
- Finding the Inverse of Toolkit Functions - Part 113:49
p258, e4
- Finding the Inverse of Toolkit Functions - Part 211:08
p258, e4
- Inverting Tabular Functions5:25
p259
- Evaluating a Function and its Inverse from a Graph at Specific Points7:54
p260, e6
- Finding Inverses of Functions Represented by Formulas - Part 113:14
p260
- Finding Inverses of Functions Represented by Formulas - Part 213:51
Examine finding inverses of a formula-based function with f(x)=x^2, x≥0, derive x=√y, then swap to get the inverse y=√x, illustrating symmetry about y=x.
- Inverting the Fahrenheit-to-Celsius Function - Example7:40
p260, e7
- Solving to Find an Inverse Function11:31
p261, e8
- Solving to Find an Inverse with Radicals11:49
p261, e9
- Finding the Inverse of a Function Using Reflection about the Identity Line9:46
p262, e10
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 this course is for:
- 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.