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K12MATH012: Precalculus I

Unit 1: Functions   Functions create a basic structure in mathematics. Some mathematicians describe them as well-functioning machines in which you can choose an input and count on the machine always producing the same output for that particular input. If a soda machine functions properly, you can input money and press cola, and the outcome is a cola. It won’t be root beer - you can count on it! A mathematical function is predictable in the same way. For example, maybe a function gives an output of 3.47 when you input the number -6. If it is truly a function, it will always give an output of 3.47 when you input the number -6. This first unit focuses on defining functions, identifying specific notation that we will be using, and helping you develop an understanding of certain characteristics of functions and their graphs. 

Unit 1 Time Advisory
Completing this unit should take approximately 19 hours and 45 minutes.

☐    Subunit 1.1: 6 hours and 10 minutes

        ☐    Subunit 1.1.1: 15 minutes

        ☐    Subunit 1.1.2: 1 hour and 25 minutes

        ☐    Subunit 1.1.3: 25 minutes

        ☐    Subunit 1.1.4: 40 minutes

        ☐    Subunit 1.1.5: 25 minutes

        ☐    Subunit 1.1.6: 3 hours and 5 minutes

☐    Subunit 1.2: 2 hours and 20 minutes

        ☐    Subunit 1.2.1: 40 minutes

        ☐    Subunit 1.2.2: 30 minutes

        ☐    Subunit 1.2.3: 1 hour and 30 minutes

☐    Subunit 1.3: 2 hours and 10 minutes

        ☐    Subunit 1.3.1: 35 minutes

        ☐    Subunit 1.3.2: 1 hour and 35 minutes

☐    Subunit 1.4: 2 hours and 30 minutes

        ☐    Subunit 1.4.1: 35 minutes

        ☐    Subunit 1.4.2: 1 hour and 55 minutes

☐    Subunit 1.5: 4 hours and 45 minutes

        ☐    Subunit 1.5.1: 1 hour

        ☐    Subunit 1.5.2: 35 minutes

        ☐    Subunit 1.5.3: 1 hour and 15 minutes

        ☐    Subunit 1.5.4: 1 hour and 55 minutes

☐    Subunit 1.6: 1 hour and 50 minutes

        ☐    Subunit 1.6.1: 1 hour and 15 minutes

Unit1 Learning Outcomes
Upon successful completion of this unit, you will be able to: - Create examples of functions and nonfunctions.
  - Describe attributes of relations that are not functions.
  - Determine the attributes of domain and range of a given function.
  - Determine whether a function is one-to-one.
  - Demonstrate facility with function notation.
  - Demonstrate facility with functions in words, tables, graphs, and formulas.
  - Execute the horizontal and vertical line tests.
  - Evaluate a function at a specific input value.
  - Solve a function given a specific output value.
  - Evaluate piecewise functions.
  - Calculate rates of change.
  - Describe basic behaviors of graphs of functions.
  - Compose functions and determine the resultant domain and range of the new function.
  - Transform functions.

Standards Addressed (Common Core):
- CCSS.Math.Content.HSA-APR.B.3 - CCSS.Math.Content.HSA-CED.4 - CCSS.Math.Content.HSA-REI.D.10 - CCSS.Math.Content.HSF-IF.A.1 - CCSS.Math.Content.HSF-IF.A.2 - CCSS.Math.Content.HSF-IF.B.4                   - CCSS.Math.Content.HSF-IF.B.5 - CCSS.Math.Content.HSF-IF.B.6                   - CCSS.Math.Content.HSF-IF.C.7 - CCSS.Math.Content.HSF-BF.A.1                 - CCSS.Math.Content.HSF-BF.B.3                 - CCSS.Math.Content.HSF-BF.B.4                 - CCSS.ELA-Literacy.RST.11-12.2                  - CCSS.ELA-Literacy.RST.11-12.3 - CCSS.ELA-Literacy.RST.11-12.4 - CCSS.ELA-Literacy.RST.11-12.5 - CCSS.ELA-Literacy.RST.11-12.7                      - CCSS.ELA-Literacy.RST.11-12.10

1.1 Functions and Function Notation   At a picnic, the number of sandwiches depends upon the number of picnickers. In a theater, the number of dollars collected depends upon the number of tickets sold. The cost of a taxi depends upon the number of miles driven. These are examples of functions, which are relations that create a unique output for any particular input. This subunit focuses on the definition of function both formally and informally.
 
The notation may be intimidating at first, but it makes later topics easier. Remember that you already think in notation. The problem, “Fifty-four minus thirty-nine,” is easier to solve when written in the notation: 54 - 39. Mathematics uses notation to simplify things and make it easier to think; it just takes time for your brain to learn new notational language.
 
Subunit 1.1 begins by defining functions and explaining a variety of ways to describe them, such as notation, graphs, and charts. It continues by introducing a set of important toolkit functions that will be extremely useful in subsequent parts of the course.

1.1.1 What Is a Function?   This subunit launches the course by introducing an important basic concept of mathematics - the function. Functions are dependable - if you know the output for a particular input, you will always get that same output for that particular input. It’s an easy idea but it can be difficult to explain.
 
Suppose you have a company building a particular toy car that has three wheels. You could write a function for wheels based on how many cars you produced. If you make one car, you need three wheels. If you make two cars, you need six wheels. If you make 300 cars, you need 900 wheels. The output depends upon the input and any one input will ALWAYS yield the same output. 

1.1.2 Function Notation   At first, function notation often seems a pointless complication, but the notation is useful and efficient for later mathematical activity. In earlier courses, you focused on an input of x and an output of y. Function notation uses the same input (x) but calls the output f(x) instead of y. This allows us to identify the name of the function f and also to identify the actual input used. For example: if f(x) = x + 2, then f(3) = 3 + 2 = 5, and f(-1) = 1. In time, it gets easier to read f(4) = 6 instead of the wordy “When x = 4, then y = 6.”

1.1.3 Solving and Evaluating Functions   This subunit gives you a chance to practice finding outputs for given functions. It is material you have covered before but probably in a little different format. It gives you more opportunity to become comfortable with function notation.

1.1.4 Graphs as Functions   Functions can be described not only algebraically but also graphically. This subsection connects these two types of function descriptions. When working with functions, it is often helpful to use both formats, because each gives a different perspective of the relation you are exploring.

1.1.5 Formulas as Functions   Think of it: If you know the diameter of a circle, you can find the circumference. It is a function because one particular diameter will ALWAYS yield one particular circumference - it won’t change. This subunit looks at relations and how to rewrite them to clarify the functional relation. For example: if x + y = 5, you can rearrange the algebra to write y = 5 - x. This would identify a function we might call g, and the algebraic description would be g(x) = 5 - x. 

1.1.6 Basic Toolkit Functions   Knowing the basic characteristics of the “toolkit functions” will be useful throughout the rest of your mathematical experiences. This subunit identifies a number of functions and helps give an overview of their behavior. Pay attention, because it will make many things much easier later on.

1.2 Domain and Range   A function takes an input, acts upon it, and produces an output. The set of inputs is called the domain, and the set of outputs is called the range. Notice that the video in the previous subunit offered a beginning look at these important characteristics of functions. Here you will reinforce those ideas, develop notation to identify the domain and range, and address how to determine domains and ranges from graphs.The final part of this subunit introduces piecewise functions, which are functions defined in pieces. 

1.2.1 Introduction and Notation   This subunit defines the terms domain and range and also develops notation for identifying sets of numbers. Remember that although notation can be confusing at first, in time it makes mathematical communication and thinking easier because it presents information in an efficient manner. 

1.2.2 Domains and Ranges from Graphs   It’s difficult to determine the domain and range of a function by looking at an algebraic representation of a function. It’s often easier to look at the graph. This subunit will help you build skills in using graphs to characterize behavior of functions. While working through the material, try to connect the algebraic and geometric representations in your mind. It will make understanding functions easier if you try to see functions in this broad perspective.

1.2.3 Piecewise Functions   Suppose bananas are $2 a pound, but if you purchase 10 pounds or more, the price is reduced to $1.50 a pound. The cost of bananas is a function of the pounds purchased. Notice, however, that there is a significant change in the function when you reach 10 pounds. This is a piecewise function, because it has to be described in pieces.

1.3 Rates of Change and Behavior of Graphs   How quickly a function changes shows a lot about a function, and the concept is a basic notion in the development of calculus. This subunit begins with average rate of change and then focuses on graphs, identifying characteristics like intervals on increase and decrease, maximums and minimums, concavity, and the trend of a function as x gets very large.

1.3.1 Rates of Change   Rate of change looks at how fast function output values change. If you travel 60 miles and it takes two hours, then your average rate of change is 30 miles per hour. This section develops skills for determining rate of change algebraically. This concept is a major concept used in calculus.

1.3.2 Behaviors of Graphs   A graph of a function shows the overall behavior, while the algebraic form indicates the specifics. In this subunit, you will identify specific behaviors of functions by investigating graphs.
 
In discussing graphs, students sometimes are confused when to identify x-values and when it is appropriate to use y-values in an answer. The x-values identify location. If a problem asks “where,” your answers will be x-values. If a problem asks the value of a function or size, then the answer will be y-values. The x coordinates indicate location, and the y-coordinates indicates size or value of the function. This subsection focuses on characteristics of graphs and discusses the characteristic behavior of the toolkit functions.

1.4 Composition of Functions   New functions are often built by putting functions together, and this subunit builds the ideas for creating those new functions. Imagine a company creating a paycheck for an employee who makes $12 per hour. A wage function can be defined as w(h) = 12h, where h is the number of hours that employee worked. Suppose further that the company has to take 20% of that amount for taxes. That would mean taking the wage and reducing it by 20%; this could be a tax function, t(w) = 0.20 w. A composition of functions makes this process easier by creating a new function by taking the output of the first function and using it as the input of the second function and then simplifying. What we want is 0.20w = 0.20(12h) = 2.4h, which shows the company taking out $2.40 for every hour the employee worked. This subunit will help you build skills to develop composed functions and restrict domains of those functions when needed.

1.4.1 Composition of Functions   The key in composition is to remember that you take the output of one function and use it as the input of another. This subunit introduces the topic and gives a significant number of examples of how to apply this important concept.

1.4.2 Composition Using Formulas   As you learned in Subunit 1.4.1, composition entails taking the output of one function and placing it into another. This subunit simplifies the process by completing the process algebraically. It’s a little confusing at first, but keep with it. It helps to write down all the pieces rather than making your head juggle too much information. 

1.5 Transformations of Functions   You will be using your toolkit functions to develop graphs of other functions. You know that the graph of f(x) = x2 is a parabola with the vertex at the origin. That function is similar to g(x) = x2 + 5. Each of the output values is 5 larger than the output values of the first function. By simply shifting the original graph up 5 units vertically, you will have the graph of the second function. This subunit helps you build an internal vision of how to see toolkit functions and use that knowledge to build graphs of more complex functions. 

1.5.1 Shifts   If you know the graph of x3, then the graph of (x-9)3 can be determined easily using a horizontal shift. The first transformation techniques you will explore are vertical and horizontal shifts in a graph. 

1.5.2 Reflections   Remember putting paint on a paper, folding it in half, and seeing the wonderful symmetric shapes you could make? This subunit looks at how reflections and symmetry of graphs can be shown in the algebraic form of a function. It also gives terms for particular types of symmetric functions. The odd functions are reflective about the y-axis [just like the graph of f(x) = x2], and the even functions are a rotation about the origin [just like the graph of g(x) = x3].

1.5.3 Stretches and Compressions   Think of having a graph of a function on a sheet of rubber. If you pull on opposite sides, you could stretch the function either horizontally or vertically. This subunit explores how an algebraic chance in a function can create a stretch or a compression of a graph.

1.5.4 Combining Transformations   Finally, you will put it all together. This subunit highlights the power of the toolkit functions and how understanding transformations can give you insight to the behavior of more complex functions. This portion of the course offers a way to visualize graphs of many functions that you will encounter throughout all further mathematics courses.

1.6 Inverse Functions   You get dressed putting on socks and then your shoes. You invert the process later by taking off the shoes first and then the socks. In this case, you start with bare feet and complete the first task. You come back later and do the task in reverse and end up where you started - with bare feet! Inverse functions are similar. If two functions are inverses, then if you take the output of one and put it in the other, you end up where you started. For example: The two functions f(x) = x + 10 and g(x) = x -10 are inverses because one will “undo” the other and give back your original input (x). Inverse functions are a major focus in precalculus, and this subunit crafts a vision of how to develop inverses and what the graphs of inverses look like.