Chapter 2. Functions and Graphs. Section 1 Functions

Transcription

1 Chapter 2 Functions and Graphs Section 1 - Functions Section 2 - Elementary Functions: Graphs & Transformations Section 3 - Quadratic Functions Section 4 - Polynomial & Rational Functions Section 5 - Exponential Functions Section 6 - Logarithmic Functions Section 1 Functions

2 Learning Objectives for Section 2.1 Functions The student will be able to do point-by-point plotting of equations in two variables. The student will be able to give and apply the definition of a function. The student will be able to identify domain and range of a function. The student will be able to use function notation. The student will be able to solve applications. 2

3 Table of Content Equations in Two Variables Definition of a Function Functions Specified by Equations Function Notation Applications 3

4 Terms point-by-point plotting function domain range independent variable dependent variable linear function constant function break-even analysis profit-loss analysis revenue function loss function profit function cost fixed/variable price-demand function 4

5 Equations in Two Variables Graphing an Equation To sketch the graph an equation in x and y, we need to find ordered pairs that solve the equation and plot the ordered pairs on a grid. This process is called point-by-point plotting. For example, let s plot the graph of the equation y = x

6 Graphing an Equation: Making a Table of Ordered Pairs Make a table of ordered pairs that satisfy the equation y = x x y 3 ( 3) 2 +2 = 11 2 ( 2) 2 +2 = 6 1 ( 1) 2 +2 = 6 0 (0) 2 +2 = 2 1 (1) 2 +2 = 3 2 (2) 2 +2 = 6 6

7 Graphing an Equation: Plotting the points Next, plot the points and connect them with a smooth curve. You may need to plot additional points to see the pattern formed. 7

8 Functions The previous graph is the graph of a function. The idea of a function is this: A correspondence between two sets D and R such that to each element of the first set, D, there corresponds one and only one element of the second set, R. The first set is called the domain, and the set of corresponding elements in the second set is called the range. For example, the cost of a pizza (C) is related to the size of the pizza. A 10 inch diameter pizza costs $9.00, while a 16 inch diameter pizza costs $

9 Function Definition You can visualize a function by the following diagram which shows a correspondence between two sets: D, the domain of the function, gives the diameter of pizzas, and R, the range of the function gives the cost of the pizza domain D range R 9

10 Function Definition TABLA 1 Dominio Rango Numero Cubo TABLA 2 Dominio Rango Numero Cuadrado Dominio Numero TABLA 3 Rango Raiz Cuadrada

11 Functions Specified by Equations If in an equation in two variables, we get exactly one output (value for the dependent variable) for each input (value for the independent variable), then the equation specifies a function. The graph of such a function is just the graph of the specifying equation. If we get more than one output for a given input, the equation does not specify a function. 11

12 Functions Specified by Equations Consider the equation that was graphed on a previous slide y = x ( 2,2) is an ordered pair of the function. 2 (-2) Input: x = 2 Process: square ( 2), then add 2 Output: result is 6 12

13 Example 2 Functions Specified by Equations Determine which of the following equations specify functions with inpependent variable x. (A) 4y 3x = 8 (B) y2 x2 = 9 13

14 Example 2 Functions Specified by Equations Solution (A) Since each input value x corresponds to exactly one output value. this equation specify a function. 14

15 Example 2 Functions Specified by Equations Solution (B) Since is always a positive real number, for any value of x, and since each positive number has two square roots, then to each input value x there corresponds two output values y. So, this equation does not specify a function. 15

16 Vertical Line Test for a Function If you have the graph of an equation, there is an easy way to determine if it is the graph of an function. It is called the vertical line test which states that: An equation specifies a function if each vertical line in the coordinate system passes through at most one point on the graph of the equation. If any vertical line passes through two or more points on the graph of an equation, then the equation does not specify a function. 16

17 Vertical Line Test for a Function (continued) This graph is not the graph of a function because you can draw a vertical line which crosses it twice. This is the graph of a function because any vertical line crosses only once. 17

18 Vertical Line Test for a Function (continued) The vertical test implies that equations of the form y = mx + b specify functions; they are called linear functions. Similarly, equations of the form y = b specify functions; they are called constant functions, and their graphs are horizontal lines. The vertical-line test implies that equations of the form x = a do not specify functions, the graph is a vertical line. y x Barnett/Ziegler/Byleen College Mathematics12e 18

19 Function Notation The following notation is used to describe functions. The variable y will now be called f (x). This is read as f of x and simply means the y coordinate of the function corresponding to a given x value. Our previous equation y = x can now be expressed as f(x) = x

20 Function Evaluation Consider our function f(x) = x What does f ( 3) mean? 20

21 Function Evaluation Consider our function f(x) = x What does f ( 3) mean? Replace x with the value 3 and evaluate the expression 2 f ( 3) ( 3) 2 The result is 11. This means that the point ( 3,11) is on the graph of the function. 21

22 Some Examples 1. f (x) 3x 2 f (2) 3(2) f ( a) 3( a) 2 f (6 h) 3(6 h) h h 22

23 Domain of a Function Consider f ( x) 3x 2 f f (0)? ( 0) 3( 0) 2 2 which is not a real number. Question: for what values of x is the function defined? 23

24 Domain of a Function Answer: f ( x) 3x 2 is defined only when the radicand (3x 2) is equal to or greater than zero. This implies that x

25 Domain of a Function (continued) Therefore, the domain of our function is the set of real numbers that are greater than or equal to 2/3. Example: Find the domain of the function 1 f ( x) x

26 Domain of a Function (continued) Therefore, the domain of our function is the set of real numbers that are greater than or equal to 2/3. Example: Find the domain of the function 1 f ( x) x 4 2 Answer: xx 8, [8, ) 26

27 Domain of a Function: Another Example Find the domain of 1 f( x) 3x 5 27

28 Domain of a Function: Another Example Find the domain of f( x) 1 3x 5 In this case, the function is defined for all values of x except where the denominator of the fraction is zero. This means all real numbers x except 5/3. 28

29 Function Notation: Example Using function notation. Find for (A) f(a) (B) f(a + h) (C) f(a + h) f(a) (D) [f(a + h) f(a)] / h h 0 29

30 Function Notation: Example Solution. (A) (B) 30

31 Solution. (C) Function Notation: Example (D) 31

32 Break-Even and Profit-Loss Analysis Applications Any manufacturing company has costs C and revenues R. The company will have a loss if R < C, will break even if R = C, and will have a profit if R > C. Costs include fixed costs such as plant overhead, etc. and variable costs, which are dependent on the number of items produced. C = a + bx (x is the number of items produced) 32

33 Break-Even and Profit-Loss Analysis (continued) Price-demand functions, usually determined by financial departments, play an important role in profit-loss analysis. p = m nx (x is the number of items than can be sold at $p per item.) The revenue function is R = (number of items sold) (price per item) = xp = x(m nx) The profit function is P = R C = x(m nx) (a + bx) 33

34 Mathematical Modeling 1 The price-demand function for a company is given by p( x) x, 0 x 100 where x represents the number of items and p(x) represents the price of the item. Determine the revenue function and find the revenue generated if 50 items are sold. 34

35 Solution Revenue = Price Quantity, so R(x)= p(x) x = (1000 5x) x When 50 items are sold, x = 50, so we will evaluate the revenue function at x = 50: R(50) (1000 5(50)) 50 37,500 The domain of the function has already been specified. We are told that 0 x

36 Example of Profit-Loss Analysis A company manufactures notebook computers. Its marketing research department has determined that the data is modeled by the price-demand function p(x) = 2,000 60x, when 1 < x < 25, (x is in thousands). What is the company s revenue function and what is its domain? 36

37 Answer to Revenue Problem Since Revenue = Price Quantity, R( x) x p( x) x ( x) 2000x 60x The domain of this function is the same as the domain of the price-demand function, which is 1 x 25 (in thousands.) 2 37

38 Profit Problem The financial department for the company in the preceding problem has established the following cost function for producing and selling x thousand notebook computers: C(x) = 4, x (x is in thousand dollars). Write a profit function for producing and selling x thousand notebook computers, and indicate the domain of this function. 38

39 Answer to Profit Problem Since Profit = Revenue Cost, and our revenue function from the preceding problem was R(x) = 2000x 60x 2, P(x) = R(x) C(x) = 2000x 60x 2 ( x) = 60x x The domain of this function is the same as the domain of the original price-demand function, 1< x < 25 (in thousands.) 5000 Thousand dollars Thousand cameras 25 39

40 Mathematical Modeling 2 Price-Demand and Revenue Modeling. A manufacturer of a popular digital camera wholesales the camera to retail outlets throughout the USA. Using statistical methods, the financial department in the company produces the price-demand data in Table A, where p is the wholesale price per camera at which x million cameras are sold. Table A. Price-Demand x (Millions) P ($) The following price-demand function was obtained: p(x) = x 1 <= x <= 15 40

41 Mathematical Modeling 2 Price-Demand and Revenue Modeling (Continued). A. Plot the data in Table A. Then sketch a graph of the pricedemand function in the same coordinate system. B. What is the company s revenue function for this camera and what is its domain? C. Complete Table B, computing revenues to the nearest million dollars. D. Plot the data in Table B. Then sketch a graph of the revenue function using this points. 41

42 Mathematical Modeling 2 Price-Demand and Revenue Modeling (Continued). Table B. Revenue x (Millions) R(x) (Million $)

43 Solution to Mathematical Modeling 2 A. y.. B. Revenue = Price Quantity, so R(x)= p(x) x = (94.8 5x) x Domain: 1 <= x <= 15.. x 43

44 Solution to Mathematical Modeling 2 C. Table B. Revenue x (Millions) R(x) (Million $) D y x 44

45 Chapter 2 Functions and Graphs Section 1 Functions END Last Update: February 14/2013