October 21, 2012 - Perth Amboy Public Schools
Unit 3 Relations & Functions pt 2.notebook October 21, 2012 A relationship between two variables can be described by sets of ordered pairs, such as (x, y) The first of the variables is called the independent variable. The second variable in the pair is called the dependent variable because it depends on the value of the first one. An example: The amount of your paycheck if you are paid hourly depends on the number of hours you worked. The independent variable is the number of hours and the dependent variable is the amount of your paycheck. Introduction to Functions Introduction to Functions Introduction to Functions Domain is the values of x that work for a given relation or equation. Relations can be expressed in several ways: A set of ordered pairs A table A mapping A graph. What is the domain for each of the following? Introduction to Functions Domain & Range Domain is the values of x that work for a given relation or equation. What is the domain for each of the following? Domain is the values of x that work for a given relation or equation. What is the domain for each of the following? xy xy 2 3 4 {2, 3, 4} xy 7 3 8 2 3 5 4 { 2, 3, 5} Domain & Range 5 8 9 1 2 {1, 2} X Y X Y X Y 1 3 1 2 3 4 2 4 2 3 1 5 5 5 3 2 0 8 3 9 4 3 1 9 4 7 5 2 3 11 {1, 2, 3, 4, 5} {1, 2, 3, 4, 5} { 3, 1, 0, 3} Domain & Range 1
Unit 3 Relations & Functions pt 2.notebook What is the domain for each of the following? What is the domain for each of the following? x y x 2 3 4 7 3 8 2 3 5 x y y 5 8 9 1 2 4 { 2, 3, 5} {2, 3, 4} October 21, 2012 {1, 2} Domain & Range Domain & Range Range is the values of y that work for a given relation or equation. We can see the domain of the relation by looking at the graph What is the range for each of the following? y 5 Since all the values of x begin at the origin and continue to the right, the domain is all real numbers greater than or equal to zero. We can write this as 4 3 2 1 x 5 4 3 2 1 0 1 1 2 3 4 5 2 3 or 4 5 Domain & Range Domain & Range Range is the values of y that work for a given relation or equation. Range is the values of y that work for a given relation or equation. What is the range for each of the following? What is the range for each of the following? x y x 2 3 4 7 3 8 2 3 5 { 3, 7, 8} y 4 {4} x y X Y X Y X Y 1 3 1 2 3 4 1 2 5 8 9 2 4 2 3 1 5 5 5 3 2 0 8 3 9 4 3 1 9 4 7 5 2 3 11 {5, 8, 9} { 5, 3, 4, 7, 9} Domain & Range {2, 3} {4, 5, 8, 9, 11} Domain & Range 2
Unit 3 Relations & Functions pt 2.notebook Describe the domain and range. October 21, 2012 Describe the domain and range. y y 5 5 4 4 3 { 2,domain 0, 2, 3} 2 2 1 1 x 5 4 3 2 1 domain All reals or 3 0 1 1 2 3 4 x 5 5 2 { 3, 2, 1, 2, 3} range 3 4 3 2 1 0 1 1 2 3 4 5 range 2 2 3 4 4 5 5 Domain & Range Domain & Range Describe the domain and range. Let's look at the graphs of functions to find the domain and range. y 5 4 domain or 3 2 1 x 5 4 3 2 1 0 1 1 2 3 4 5 2 range or 3 4 5 Link to interactive grapher tiFunctionDataFly/ Domain & Range A relation is discrete if it is made up separate points. For example, you go to a bakery to buy donuts. How many can you buy? 0, 1, 2, 3, . . . These are separate values. 1.2, 1.375, 1.5899 do not have meaning. A relation is continuous if the points are not separate. Domain & Range Are the following relations discrete or continuous? x y 2 3 4 7 3 8 For example, the repairman says he will be to your home between 1 and 5. What time could he show up? 1, 2, 3, 4, 5. Do the values between 1 and 2 have meaning? Discrete v. Continuous X Y 1 3 2 4 5 5 3 9 4 7 Discrete v. Continuous 3
Unit 3 Relations & Functions pt 2.notebook 1 Is the following relation discrete or continuous? October 21, 2012 2 Is the following relation discrete or continuous? y 5 4 A Discrete y 5 3 A Discrete 2 4 3 1 x B Continuous 5 4 3 2 1 0 1 1 2 3 4 2 5 1 x B Continuous 2 5 4 3 2 1 0 1 1 2 3 4 5 3 2 4 3 5 4 5 Discrete v. Continuous Discrete v. Continuous 3 Is the following relation discrete or continuous? A function is a relation that has each value in the domain has exactly one value in the range. In other words, x does not repeat. A Discrete Which of the following relations is a function? B Continuous Discrete v. Continuous Relation v. Function 4 Which of the following relations is a function? Which of the following relations is a function? x y x 2 3 4 7 3 8 2 3 5 X Y X Y X Y 1 3 1 2 3 4 y x y 2 4 2 3 1 5 5 8 9 5 5 3 2 0 8 4 1 2 3 9 4 3 1 9 4 7 5 2 3 11 A Relation v. Function B C Relation v. Function 4
Unit 3 Relations & Functions pt 2.notebook October 21, 2012 On a graph, a function does not have a point directlyabove another point. This is because there cannot be 2 y values for the same x value. This is called the Vertical Line Test. Graph B represents a function, but A and C do not. y 5 5 4 4 4 3 3 3 2 2 1 5 4 3 2 1 0 1 1 2 3 4 5 1 x 5 4 3 2 1 0 1 1 2 3 4 5 x 5 4 3 2 1 0 1 2 2 2 3 3 3 4 4 4 5 5 5 A Yes No 2 1 x 5 Is the following relation a function? y y 5 B 1 2 3 4 Yes No {(3,1), (2, 1), (1,1)} y 2 1 0 5 C Relation v. Function 6 Is the following relation a function? x 1 0 3 Relation v. Function 7 Is the following relation a function? Yes No Relation v. Function X Y 2 3 0 2 1 1 3 2 2 0 Relation v. Function 9 Is the following relation a function? 8 Is the following relation a function? Yes No Yes No Relation v. Function Relation v. Function 5
Unit 3 Relations & Functions pt 2.notebook 10 Is the following relation a function? October 21, 2012 11 Is the following relation a function? Yes Yes No No Relation v. Function 12 Is the following relation a function? Yes No Relation v. Function 13 Is the following relation a function? Yes No Relation v. Function When a function f is defined with a rule or an equation using x as the independent variable and y as the dependent variable, we say y is a function of x and use the notation y f(x) . Note: f(x) does not mean f times x, but f(x) is read as "f of x ". Example: If we have a function y 2x- 5, we just replace y with f(x) and the equation is written f(x) 2x-5. Function notation Relation v. Function When we want to find the value of y for a given value of x, we substitute the value of x into the equation. Example: We want to find y 2x-5 when x 3. In function notation, the equation is written as f(x) 2x- 5. We use f(3) to indicate we are replacing x by 3 to find y. f(3) 2(3)-5 6-5 1. Remember, f(x) is the same as y, and f(a) is the value of y when x a . Function notation 6
Unit 3 Relations & Functions pt 2.notebook 14 Find f(2) if October 21, 2012 15 Find and simplify g(a 1) if A B C D 2a 3 2a 4 2a 5 5 Function notation 16 Find f(3) Function notation 17 Find f(‑1) -2 3 10 6 5 12 Function notation If a function is not in "y " form, we have to solve the equation for y. Example: x - 4y 5 Solving for y, we get Function notation 18 Find f(‑3) for which is the same as So And f(-2) would be Function notation Function notation 7
Unit 3 Relations & Functions pt 2.notebook 19 What is the missing value in the table? x f(x) 5-x2 0 5 1 4 -3 October 21, 2012 A linear function is defined as f(x) mx b When m 0, the linear function becomes f(x) b and it is called a constant function. An example of a constant function is f(x) 5. The graph of a constant function is a horizontal line. The domain of a linear function is numbers, or (- , ). all real The range of a linear function is all real numbers , or (- , ), except that the range of a constant function is equal to the constant, b. Function notation Function notation Oct 21 4:27 PM 8
Unit 3 Relations & Functions pt 2.notebook 3 October 21, 2012 Domain & Range Describe the domain and range. 5 4 3 2 1012345 5
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