Functions Review Today’s Agenda Do Now Questions Review LT: I will study to be ready for my assessment   Today’s Agenda Do Now Questions Review Success Criteria II can prepare for my assessment

Clear your desk including back packs Put away your phones Functions Test LT: I will add, subtract and multiply functions. Describe the domain and range and describe key features of a function. Test: Clear your desk including back packs Put away your phones Make sure you have a pencil and eraser When finished put your test in the box Two sheets of hand written notes are permitted Please -no talking during the Test Remain quiet until everyone is finished Today’s Agenda Test Success Criteria II can add, subtract and multiply functions. Describe the domain and range and describe key features of a function.

Review: What is a function? A relationship where every domain (x value has exactly one unique range (y value). Sometimes we talk about a FUNCTION MACHINE, where a rule is applied to each input of x

Domain and Range x x Solve for x if i(x)=12 Do Now Lesson LT: I will be able to evaluate functions and identify key concepts from graph, table, or algebraically.     x f(x) x f(x)     Solve for x if i(x)=12   Today’s Agenda Do Now Lesson Class activity HW#8 Success Criteria I can evaluate functions and identify key concepts from the graph, table , or algebraically

Pull out HW# 7

Recreate the graph on your poster Describe the x-axis and y-axis Are there any x and y intercepts and how do you know ( what if x=0 and what if y=0)-describe it Describe the Domain in your own words How do you know Use set builder notation and interval notation to describe the domain Describe the Range in your own words Use set builder notation and interval notation to describe the Range Be prepared to describe your poster to the class Task Master Artist Presenter Re-voicer

HW# 8 Work in pairs-Not Groups

Hand out Activity

Adding and Subtracting functions LT: I will learn that functions can be added and subtracted.   Today’s Agenda Do Now Activity Lesson HW#9 Success Criteria I can add and subtract functions

Adding and Subtracting functions LT: I will learn that functions can be added and subtracted. Today’s Agenda Do Now Lesson HW#10 Success Criteria I can add and subtract functions

Function Operations

Combine like terms & put in descending order The sum f + g This just says that to find the sum of two functions, add them together. You should simplify by finding like terms. Combine like terms & put in descending order

Adding Linear Functions f(x) + g(x) If f(x)= 3x - 4 and g(x) = -2x + 6, find f(x) + g(x). (3x – 4) + (-2x + 6) OR 3x – 4 + -2x +6 1x + 2 So, f(x) + g(x) = 1x + 2

Adding Linear Functions If f(x) = 7x + 3 and g(x) = 6x – 4, find f(x) + g(x).

The difference f - g Distribute negative To find the difference between two functions, subtract the first from the second. CAUTION: Make sure you distribute the – to each term of the second function. You should simplify by combining like terms. Distribute negative

Adding and Subtracting Functions When we look at functions we also want to look at their domains (valid x values). In this case, the domain is all real numbers.

Subtracting Linear Functions f(x) - g(x) If f(x) = 4x -29 and g(x)= 2x – 18, find f(x) – g(x). (4x – 29) – (2x -18) change – to + the opposite. (4x – 29) + (-2x +18) Or 4x – 29 + -2x + 18 2x - 11 So, f(x) – g(x) = 2x – 11

Subtracting Linear Functions If f(x) = -3x + 15 and g(x) = -4x -17, find f(x) – g(x).

HW#9

Adding and Subtracting functions I will learn that functions can be added and subtracted. LT: I will be able to evaluate functions and identify key concepts from graph, table, or algebraically. What is the domain What is the range At what intervals is the function increasing At what intervals is the function decreasing Where are the x intercepts Where are the y intercepts Today’s Agenda Do Now Review Catch up day Quiz tomorrow Success Criteria I can add and subtract functions I can evaluate functions and identify key concepts form a graph, table, or algebraically

Adding and Subtracting functions And Domain and Range I will learn that functions can be added and subtracted. LT: I will be able to evaluate functions and identify key concepts from graph, table, or algebraically. Quiz: Clear your desk including back packs Put away your phones Make sure you have a pencil and eraser When finished, staple your quiz and put it in the box HW#8, 9 & 10 are the only notes permitted No talking during the quiz and remain quiet until everyone is finished Today’s Agenda Quiz Work on unfinished assignments Success Criteria I can add and subtract functions I can evaluate functions and identify key concepts form a graph, table, or algebraically

Multiplying Functions I will multiply functions using different methods Do Now 1. 33 27 2. 4 • 4 • 4 • 4 256 3. b2 for b = 4 16 4. n2r for n = 3 and r = 2 18 Today’s Agenda Do Now Hand Back Quiz Lesson HW#11 Success Criteria I can multiply functions using different methods

Multiplying Polynomials

The Distributive Property (x + 5)(2x + 6) (x+5) 2x + (x+5)6 2x(x+5) +6(x+5) 2x² + 10x + 6x + 30 2x² + 16x +30 NOTE : Since there are THREE terms this is called a TRINOMIAL

Area Model (4x+3)(2x+9) 4x 3 2x 9            

Trinomials Multiplying MOST binomials results in THREE terms You can learn to multiply binomials in your head by using a method called F O I L

first terms last terms (x + 4) ( x + 2) inner terms outer terms The FOIL Method (x + 4)(x +2) first terms last terms (x + 4) ( x + 2) inner terms outer terms

terms terms terms terms (x + 4) ( x + 2) Now write the products x² + 2x + 4 x + 8 first outer inner last terms terms terms terms

multiply the first two terms multiply the two outer terms To Multiply any two binomials and write the result as a TRINOMIAL follow these steps multiply the first two terms multiply the two outer terms multiply the two inner terms multiply the last two terms

Using FOIL k ² + 4k - 4k - 16 k ² - 16 (6c – 3) ( 6c + 3) A Closer Look (k – 4) (k + 4) Using FOIL k ² + 4k - 4k - 16 k ² - 16 (6c – 3) ( 6c + 3) 36c² +18c – 18c – 9

Special Cases Difference of Squares Perfect Square Trinomials There are several special cases of multiplying binomials Difference of Squares Perfect Square Trinomials

Difference of two squares When you multiply the sum of two terms and the difference of two terms you get a BINOMIAL (a + b) (a – b) = a² - b² This binomial is the difference of two squares

Good idea to put in descending order but not required. The product f • g To find the product of two functions, put parenthesis around them and multiply each term from the first function to each term of the second function. FOIL Good idea to put in descending order but not required.

Multiplying Functions In this case, the domain is all real numbers because there are no values that will make the function invalid.

Dividing Functions In this case, the domain is all real numbers EXCEPT -1, because x=-1 would give a zero in the denominator.

Nothing more you could do here. (If you can reduce these you should). The quotient f /g To find the quotient of two functions, put the first one over the second. Nothing more you could do here. (If you can reduce these you should).

Let’s Try Some What is the domain?

Perform each operation                 Pg. 401 #9-75 by 3s    

Let’s Try Some What is the domain?

Let’s Try Some What is the domain?

Let’s Try Some What is the domain?

Composite Function – When you combine two or more functions The composition of function g with function is written as 1 1. Evaluate the inner function f(x) first. 2. Then use your answer as the input of the outer function g(x). 2

Example – Composition of Functions Method 1: Method 2:

Let’s try some

Solution

Solving with a Graphing Calculator Start with the y= list. Input x3 for Y1 and x2+7 for Y2 Now go back to the home screen. Press VARS, YVARS and select 1. You will get the list of functions. Using VARS and YVARS enter the function as Y2(Y1(2). You should get 71 as a solution.

Real Life Application You are shopping in a store that is offering 20% off everything. You also have a coupon for $5 off any item. Write functions for the two situations. Let x = original price. 20% discount: f(x) = x – 0.20x = 0.8x Cost with the coupon: g(x) = x - 5

2. Make a composition of functions: You are shopping in a store that is offering 20% off everything. You also have a coupon for $5 off any item. 2. Make a composition of functions: This represents if they clerk does the discount first, then takes $5 off the discounted price.

3. Now try applying the $5 coupon first, then taking 20% off: You are shopping in a store that is offering 20% off everything. You also have a coupon for $5 off any item. 3. Now try applying the $5 coupon first, then taking 20% off: How much more will it be if the clerk applies the coupon BEFORE the discount?

4. Subtract the two functions: You are shopping in a store that is offering 20% off everything. You also have a coupon for $5 off any item. 4. Subtract the two functions: Any item will be $1 more if the coupon is applied first. You will save $1 if you take the discount, then use the coupon.