Relation: Pairs of items that are related in a predictable way.

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1 We begin this unit on a Friday, after a quiz. We may or may not go through these ideas in class. Note that there are links to Kahn Academy lessons on my website. Objective 1. Recognize a relation vs. a function. Identify the input & output of a function A: #1 (Functions & relations) Relation: Pairs of items that are related in a predictable way. Eample: You have $100 in a bank account. You make a deposit or withdrawal. There is a relation between the balance in the bank and the transaction. Relations have an " input" and an "output" Input (Transaction) Output (Balance) (0, 10), ( 10, 90), (30, 130), ( 7, ) The set of all possible "inputs" is called the "Domain" of the relation. The set of all possible "outputs" is called the "Range" of the relation. A function is a relation that has eactly one output for every input Snake Dog Cat 0 Is this a function? 30 Bird Is this a function? Animals (Not so Healthy) Snake # of Legs 0 7 Dog Human Bird Are these functions? (, 6), (, ), (3, 6), (, ) (1, 6), (, ), (1, ), (, ) Determining whether a relation is a function is easy from a graph: Vertical Line Test: If a vertical line passes through the graph of a relation eactly once for all in its domain, then the relation is a function in that domain. A: #1 (Functions & relations) 9/ A: Relations 1

2 Hand out Unit Plans! Note that there is a lot of support at Kahn Academy for this. A: #1 (Functions & relations) Discuss. Go back to lesson if needed Objective 1. Interpret and use function notation B: #ace,all,6,9,11 (Function notation) C: #1adghi,,3all,jkl (Domain & Range) Most often we use an algebraic rule to represent a relationship (or function). It's important to understand and use different notations. Here are three equivalent forms: y = 100 f() = 100 Notation: If f() = 100, then f() = 100 = 10 Think of a "function" machine that performs some operation: f Times 3 7 f() f() = 3 7 f() = 3() 7 = 1 7 = 19 g Square & add 3 g() g() = 3 g( ) = ( ) 3 = 3 = 8 The output value is sometimes called the image of. 9/6 B: Function Notation

3 Objective 1. Understand, find and use the domain and range of a function from a graph or from an algebraic form of a function. Domain: The set of all possible values of the input ( ) to a relation. Range: The set of all possible values of the output ( y) of a relation. Graphical Notation: Open circle means the point is not included. Solid circle means the point is included. Arrow head means that the curve continues indefinitely in that direction. Domains and ranges are sets so we use set notation to describe them We also need to look at finding the domain and range just from the algebraic form of the function without graphing it. Domain and Range can be trickier to find from the algebraic form of a function. But there are general guidelines: Domain: Look for values of that cause: A zero in a denominator (can't do that Adolf) A negative number inside a square root (only in your dreams) A non positive argument to a log function (we'll get to that a bit later) Range: Consider the value of the function as gets very large in both directions at the endpoints of the domain as a general shape. For eample, > Even order polynomials have maima or minima > Odd order polynomials have an infinite range > Eponentials will always be positive unless shifted vertically > Sin and cos have a restricted range. Sometimes rewriting the function in another form will help you see the domain and range. Up until now, we have usually been incomplete when we write a function because we do not define the domain. We have assumed the natural domain which is the largest set of 's in for which the function eists. B: #ace,all,6,9,11 (Function notation) C: #1adghi,,3all,jkl (Domain & Range) 9/6 C: Domain and Range 3

4 B: #ace,all,6,9,11 (Function notation) Present e,6,11 C: #1adghi,,3all,jkl (Domain & Range) Present 1i, 3visual, jl 1. Create composite functions of a variable. Evaluate composite functions for specific values of. 3. Find the domain and range of composite functions. D: # 6 (Composite functions) g f f() g(f()) = (g f)() Square Times 3 7 f() = 3 7 & (g f)() = (3 7) 3 add 3 Eample: f() = 3 and g() =. Find (f g)() Find (f g)() Find (g f)() Find (g f)() The composite of two functions is created by using the output of one function as the input to the other function. Try: Given f() = 7 and g() = find (f g)() and (g f)() (f g)() = ( ) 7 (g f)() = ( 7) Let's look at the Domain and Range more carefully Domain of f Range of f Domain of g Range of g f f() g g(f()) = (g f)() Times 3 7 f() = 3 7 Square root (g f)() = Domain of f() is: ], [ Domain of g() is: { 7/3} Domain of (g f)() is: { 7/3} Finding the domain and range of composite functions requires analysis of all constituent functions! Properties of Composite Functions (f g)() is not the same as ( g f)() in general The range of the first function in a composition is the domain of the second. The domain and range of a composite function depend on the domains and ranges of all the constituent functions. D: # 6 (Composite functions) 9/7 D: Composite Functions

5 F Block only: Present D:,3c,,,6 (if time) E: #1 last col (Sign diagrams (use technology to confirm only)) QB: #1,a,3,b,6a,8b (IB Practice) 1. Draw and interpret sign diagrams - from graphs and from functions. Sometimes it's helpful to visualize a function. Short of creating a full graph of it, one can create a sign diagram showing just the sign of the function over its domain. The value of a function can only change sign at a place where a) the graph of the function passes through zero or b) the graph of the function has a vertical asymptote. The horizontal line represents the ais The critical values are written below the horizontal line. They are the intercepts and vertical asymptotes of the function. The sign of the function ( or ) is written between critical values above the horizontal line. Notice that the sign does not always change between critical values. Factoring (writing as a product) is a very helpful tool in creating sign diagrams because if any factor is zero, then the whole epression is zero! Notice how vertical asymptotes are represented. A vertical asymptote will occur at any value of that causes the function to divide by zero. Draw sign diagrams for: 9 (3 )( ) ( 1)( ) Some hints for the QB questions (reminders from MYP ): QB #1: ln() is only defined for > 0 QB #3: involves the cosine function. Write the function, solve it if you can. You will see some functions that you haven't seen for a while. Don't panic. E: #1 last col (Sign diagrams (use technology to confirm only)) QB: #1,a,3,b,6a,8b (IB Practice) 9/9 E: Sign Diagrams

6 Quiz on Chapter this Thursday, 10/6 E: #1 last col (Sign diagrams (use technology to confirm only)) Present lo,dhi QB: #1,a,3*,b,6a,8b (IB Practice) Present all F: #1 (Rational Functions) Also see Kahn Academy Videos Google "Kahn Rational Function". Links on website. 1. Understand, identify the key features of, and sketch rational functions (linear/linear) Rational Functions and Asymptotes The function is called the Reciprocal Function. By plotting a few points, try to graph it. What do you think the following graphs would look like? Reciprocals A line that a function approaches but never reaches is called an asymptote. They can be horizontal, vertical, or skew (oblique, slant). The notation we use for describing asymptotes is as follows: Some other features to notice: The shape is called a rectangular hyperbola The function is undefined when = 0 The function only eists in the first and third quadrants The and y aes are both asymptotes More general rational functions are those that can be written as a ratio of two polynomials. Rational Functions are functions that can be written as a ratio of two polynomials. That is where p and q are polynomials. Rational functions may have: Vertical asymptotes: at values of where the denominator is zero. Zeros or roots: at values of where the numerator is zero. Horizontal asymptotes: a y value that the function approaches as gets very large or small. For this course, we will restrict ourselves to rational functions that are a ratio of two linear functions. That is, functions of the form Let's look at these more closely. Consider the function Vertical asymptote? Horizontal asymptote? Features of rational functions of the form The shape is a rectangular hyperbola The function has a vertical asymptote at = d/c The function has an horizontal asymptote at f() = a/c The function has a zero at ( b/a, 0) The function has a y intercept at (0, b/d) #c is a challenge! Don't stress over it if you see no path... F: #1 (Rational Functions) Also see Kahn Academy Videos Google "Kahn Rational Function". Links on website. Quiz on Chapter this Thursday, 10/6 10/3 F: Rational Functions 6

7 Quiz on Chapter this Thursday, 10/6 F: #1 (Rational Functions) Present F: 1abcd, why?, c ideas 1. Understand the relationship between a function and its inverse.. Know how to find an inverse function algebraically and graphically. What is the inverse of addition? multiplication? eponentiation? subtraction? division? taking roots? Inverses undo an operation. An inverse function undoes the operation of a forward function. Eample: f() = 3 multiplies by 3, then adds two G: #,,,7,8,10,1,1 (Inverse functions) QB: #bc,,a,6b,7,8a,9,10,11 (IB Practice) 3. Recognize domain and range constraints and relationships for inverse functions Pick a number. Double it. Add. Write down your result. Start with your result. Subtract. Divide by two. Write down your result. It's the original number! (Rocket science 101!) The inverse function, called has to subtract then divide by 3 so What is the inverse function of f() =? How do we find an inverse function in general? Graphically Graph the reflection over y = Algebraically Switch the and y, then solve for y. Try finding the inverse of y = 3 Graphically Algebraically In general if you perform some rule on then perform the inverse rule, you get. of f Domain Range of f f f() Domain of Range of In mathematical notation: f(f 1 ()) = and also f 1 (f()) = or Notice from the above figure that The range of f defines the domain of f 1 The range of f 1 defines the domain of f The domain of f 1 is the range of f The domain of f is the range of f 1 Some properties of inverse functions: The inverse of f() must itself be a function (satisfies vertical line test) The inverse of f() is the reflection of f() over the line y =! A self inverse function is it's own inverse! Any function that is symmetric around y = is a self inverse. Eample: Find and graph the inverse of Describe the domain and range of the function and its inverse. 10/: G:Inverse Functions G: #,,,7,8,10,1,1 (Inverse functions) QB: #bc,,a,6b,7,8a,9,10,11 (IB Practice) 7