1. CHAPTER 1: FUNCTIONS, GRAPHS, AND MODELS; LINEAR FUNCTIONS Section 1.1: Functions and Models 1

2. SECTION 1.1: FUNCTIONS AND MODELS A function is a rule that assigns to each element of one set (called the domain) exactly one element of a second set (called the range). How can we express functions? Set of ordered pairs: {(-20, -4) (-10, 14) (-5, 23) (0, 32) (25, 77) (50, 122) (100, 212)} A table: 2 Celsius (˚C) -20-10-502550100 Fahrenheit (°F) -414233277122212

3. SECTION 1.1: FUNCTIONS AND MODELS How can we express functions? A graph: An equation: 3

4. SECTION 1.1: FUNCTIONS AND MODELS ‘ Parts ’ of a Function Domain: set of all input values (usually x) x is called the independent variable x is what the function ‘ acts upon ’ Range: set of all output values (usually y) y is called the dependent variable y is the result of the function acting on the independent var. x, input  function, f  f(x) = y, output f(x) is read as “ f of x ” which indicates that the function f, is acting upon the variable, x 4

5. SECTION 1.1: FUNCTIONS AND MODELS Determine whether each example represents a function. 5 x, Year199119921993199419952000 N, Worldwide PCs (millions)129.4150.8177.4208.0245.0535.6

6. SECTION 1.1: FUNCTIONS AND MODELS Determine whether each example represents a function. W is a person ’ s weight in pounds during the nth week of a diet for n = 1 and n = 2. y 2 = 3x - 3 6

7. SECTION 1.1: FUNCTIONS AND MODELS What about a graph? How can we determine if a graph is a function? y 2 = 3x - 3 y = -x 2 + 4x Vertical Line Test – a set of points in a coordinate plane is the graph of a function if and only if no vertical line intersects the graph in more than one point. 7

8. SECTION 1.1: FUNCTIONS AND MODELS Functional Notation: y = f(x) y = f(x) is the output for some value of x The point (a, f(a)) lies on the graph of y = f(x). Find f(3) and f(-1) if f(x) = 4x 2 - 2x + 3 If Fahrenheit is a function of Celsius, F(C), find F(50) and F(-10). 8 Celsius (˚C) -20-10-502550100 Fahrenheit (°F) -414233277122212

9. SECTION 1.1: FUNCTIONS AND MODELS The points on the graph give the number of men in the workforce (in millions) as a function g of the year for selected years t from 1890 to 1990. 9

10. SECTION 1.1: FUNCTIONS AND MODELS Find and interpret g(1940). What is the input t if the output is g(t) = 51.6 million men? What can you say about the number of men in the workforce during 1890-1990? 10

11. SECTION 1.1: FUNCTIONS AND MODELS General Electric ’ s revenue in billions of dollars can be represented by R(x) = 10.03x – 19935.19 where x represents the year from 1994 through 1998. Find GE ’ s revenue for 1996 and 1998. 11

12. SECTION 1.1: FUNCTIONS AND MODELS Domains and Ranges When we look for domains, we are looking for the values that we are ‘ allowed ’ to plug in. (or what values will pose a problem?) In general, we assume that the domain is all real numbers, unless we get a 0 in the denominator we are taking the even root (e.g. square root) of a negative number The range is simply the function output of the specified domain. 12

13. SECTION 1.1: FUNCTIONS AND MODELS Find the domain for each of the following: y = 4x 2 y = 13

14. SECTION 1.1: FUNCTIONS AND MODELS Homework: pp. 19-25 1-23 odd, 29, 33, 37, 41, 45, 53, 57 14