If r = 9, b = 5, and g = −6, what does (r + b − g)(b + g) equal? Do Now If r = 9, b = 5, and g = −6, what does (r + b − g)(b + g) equal? −20 −8 8 19 20 ACT 2014 Exam 2.1.3: Domain and Range

GOOD THINGS!! 2.1.3: Domain and Range

Note Taking MUST include: Today’s date (9/2) before you begin notes Essential Question at top of page Essential Questions HOW IS A FUNCTION DEFINED? HOW DO THE DOMAIN AND RANGE OF A FUNCTION RELATE TO POINTS IN THE COORDINATE PLANE? 2.1.3: Domain and Range

Warm Up DO NOT WRITE IN THE WORKBOOKS Turn to page 7 Answer the questions 1-3 in your notes for today DO NOT WRITE IN THE NOTEBOOKS We will discuss answers as a class in ~4 minutes 2.1.3: Domain and Range

Warm Up Review What is the gate’s input? Children under age 10 that do not have caps What is the gate’s output? Children with caps What might be implied if all the rules were followed but there were still children 10 years old and younger at the ballpark without caps? We can imply that they did not go through Gate 7 2.1.3: Domain and Range

Note Taking Strategies You may take notes however you like – you MUST put the date and essential question at the top of your notes for the day If you see a that’s my hint to you to put that in your notes Write things down in your own words You remember 90% of what you teach Writing it in your own words can help you teach it to someone else 2.1.3: Domain and Range

Introduction – Relation A relation is any relationship between two sets of data If you pair any two sets of data, you create a relation between the sets You could define a relation between the neighbors on your block and the cars they drive If you are given a neighbor’s name, you can properly assign that person to his or her car 2.1.3: Domain and Range

Introduction, continued A function is a special relation in which each input is mapped to only one output. So, as with the first example, say we are given a name of a neighbor and we assign that neighbor to a make and model of a car. Even if some of your neighbors drive the same make and model car, each person’s name is assigned to only one type of car. This relation is a function. 2.1.3: Domain and Range

Introduction, continued But, what if we go the other way and look at the make and model of the cars and map them to the neighbors who drive them? Then a particular make and model of a car may be assigned to two or more neighbors. This is NOT a function. 2.1.3: Domain and Range

Relations 2.1.3: Domain and Range Specify to the students that the first relation is a function because there is only one range for each domain, but the second is not a function because there are multiple ranges for the same domain. The domain represents neighbors on the street and the range represents the types of cars. 2.1.3: Domain and Range

Domain and Range Input = domain of the function Output = range of the function A function f takes an element x from the domain and creates f(x), an element in the range f(x) is strictly one value Each element in the domain of a function can be mapped to exactly one element in the range For every value of x, there is exactly one value of f(x). 2.1.3: Domain and Range

Functions – Cake Example Water Sugar Eggs Flour Function OUTPUT Each input only has one output – if you put these eggs in the oven, you will never get a soda. The only possible output is cake. INPUT 2.1.3: Domain and Range

Functions, cont. 2 14 8 3 19 74 5 6 46 35 f(x) = x2 +10 Just like the cake ingredients, these inputs go through a function and we get an output 2 14 8 3 19 74 5 6 f(x) = x2 +10 46 35 Function Domain Range If you put the number 2 in this function, you will ONLY ever get the number 14. This input has EXACTLY ONE output. 2.1.3: Domain and Range

Domain and Range X domain input y range output All of these come before in the alphabet  X domain input y range output 2.1.3: Domain and Range

Vertical Line Test If a vertical line crosses a graph at two unique points, this means that the graphed relation has two unique values for f(x) But we know that a relation is only a function if f(x) has one unique value If a vertical line crosses the graph in only one place, the graph is a function If the line crosses the graph in two or more places, it is not a function. 2.1.3: Domain and Range

Function or not a function? 2.1.3: Domain and Range

Why does the vertical line test work????? What do you know about functions? What does a line passing a graph at two points mean? Think in terms of domain and range If the line crosses two points, then that means that there are two outputs for that input 2.1.3: Domain and Range

Defend the statements below: Is not a function: Is a function: 2.1.3: Domain and Range

Key Concepts The domain is the set of x-values that are valid for the function. The range is the set of y-values that are valid for the function. A function maps elements from the domain of the function to the range of the function. Each x in the domain of a function can be mapped to one f(x) in the range only. 2.1.3: Domain and Range

Key Concepts, continued The relation is not a function if an element in the domain maps to more than one element in the range To create a mapping, list the domain in one column and the range in a second column Draw lines that match the elements in the domain to the corresponding elements in the range In the mapping, if one line goes from one x-value to only one y-value, then the relation is a function. 2.1.3: Domain and Range

Key Concepts, continued Vertical line test If you pass an imaginary vertical line across the graph, look to see if the line ever crosses more than one point on the graph at a time 2.1.3: Domain and Range

Common Errors/Misconceptions confusing domain and range thinking that if a range value repeats or is the same for a different value in the domain that the relation is not a function (different x-values have the same y-value) 2.1.3: Domain and Range

Partner Practice Page 9-11 You have ~20 mins to complete as many problems as you can with your elbow partner DO NOT WRITE IN THE WORKBOOKS Explain your reasoning in your spiral notebooks – this will help you when you go back to study 2.1.3: Domain and Range

Identify the Domain and Range of the Graph Note the arrows! (what does this mean?) Domain: {all real numbers} Range: {y > 0} 2.1.3: Domain and Range

Independent Practice Pages 15-18 2.1.3: Domain and Range

Exit Ticket Must be done independently before the bell rings When you are done, quietly finish the problems you did not get to earlier with your partner 2.1.3: Domain and Range

Khan Academy Class code: KEBJUF Once you are added to the class, look for “Manipulating Functions” Get started on “Intro to combining functions” 2.1.3: Domain and Range