Chapter 2 Functions and Graphs Section 1 Functions

2 Barnett/Ziegler/Byleen Business Calculus 12e 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.

3 Graphing Equations  If you are not familiar with a graph’s “family”, then use point-by-point plotting. (i.e. make an x-y table) However, this is a very tedious process.  Knowing a graph’s family, will help you determine its basic shape.  Knowing a graph’s basic shape and the transformations on its parent, will help you graph it without making an x-y table. This will be reviewed in tomorrow’s lesson. Barnett/Ziegler/Byleen Business Calculus 12e

4 Families and Shapes Barnett/Ziegler/Byleen Business Calculus 12e

5 Families and Shapes Barnett/Ziegler/Byleen Business Calculus 12e

6 Functions  A relation (set of ordered pairs) represents a function if for each x, there is only one y.  The set of all x’s is called the domain, and the set of all corresponding y’s is called the range.  Which of these relations is a function? {(1, 3), (4, 9), (7, 15), (10, 21)} {(2, 4), (-2, 4), (3, 9), (-3, 0)} {(16, 4), (16, -4), (9, 3), (9, -3)} Answer: The first two are functions.

7 Barnett/Ziegler/Byleen Business Calculus 12e Vertical Line Test for a Function If you have the graph of an equation, you can easily determine if it is the graph of a function by doing the vertical line test.

8 Barnett/Ziegler/Byleen Business Calculus 12e Vertical Line Test for a Function (continued) This graph fails the vertical line test, so it’s not a function. This graph passes the vertical line test, so it is a function.

9 Barnett/Ziegler/Byleen Business Calculus 12e 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. can now be expressed as

10 Barnett/Ziegler/Byleen Business Calculus 12e Function Evaluation  Consider our function  Evaluate: f (–3) (-3) 2 – 2 = 7 f(a) a f(2x) (2x) 2 – 2 = 4x 2 – 2 f(x + h) (x + h) 2 – 2 = x 2 + 2xh + h 2 – 2

11 Barnett/Ziegler/Byleen Business Calculus 12e More Examples

12 Barnett/Ziegler/Byleen Business Calculus 12e Domain of a Function  The domain of a function refers to all the possible values of x that produce a valid y.  The domain can be determined from the equation of the function or from its graph.

13 Barnett/Ziegler/Byleen Business Calculus 12e Finding the Domain of a Function If a function does not contain a square root or a denominator then its domain is all reals (- ,  )

14 Determining Domain  If a function contains a square root or a denominator containing x, its domain will be restricted.  The next few examples show how to determine the restricted domain. Barnett/Ziegler/Byleen Business Calculus 12e

15 Barnett/Ziegler/Byleen Business Calculus 12e Finding the Domain of a Function Functions with square roots: Set the expression inside the square root  0 and solve for x to determine the domain.

16 Barnett/Ziegler/Byleen Business Calculus 12e Finding the Domain of a Function  Example: Find the domain of the function

17 Barnett/Ziegler/Byleen Business Calculus 12e Finding the Domain of a Function  Functions with x in the denominator: Set the denominator  0 and solve for x to determine what x cannot be equal to.

18 Barnett/Ziegler/Byleen Business Calculus 12e Finding the Domain of a Function Find each domain:

19 Business Analysis  Types of relations involving business applications: Total Costs = fixed costs + variable costs C = a + bx(linear relation) Price-Demand function = the price for which an item should be sold when you know the demand p = m – nx(linear relation) Price-Supply function (similar to above) Revenue = number of items sold  price per item R = x  p = x(m – nx)(quadratic relation) Profit = Revenue – Cost P = x(m – nx) – (a + bx) (quadratic relation) Barnett/Ziegler/Byleen Business Calculus 12e

20 Barnett/Ziegler/Byleen Business Calculus 12e Mathematical Modeling The price-demand function for a company is given by where x represents the number of items and p(x) represents the price of the item. A) Determine the revenue function. B) Find the revenue generated if 50 items are sold. C) What is the domain of the revenue function?

21 Barnett/Ziegler/Byleen Business Calculus 12e Solution A) Revenue = Quantity  Price R(x) = x ∙ p = x(1000 – 5x) R(x) = 1000x – 5x 2 B) When 50 items are sold, we set x = 50: C) The domain of the function is the same as the domain for the price-demand function (which was given):

22 Barnett/Ziegler/Byleen Business Calculus 12e Break-Even and Profit-Loss Analysis  Any manufacturing company has costs C and revenues R.  They determine the following: If R < C  loss  If R = C  break even  If R > C  profit

23 Barnett/Ziegler/Byleen Business Calculus 12e 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, p(x) is in dollars). A) What is the price per computer when the demand is 20 thousand computers? B) What is the company’s revenue function and what is its domain? C) How much revenue is generated for 20 thousand computers?

24 Barnett/Ziegler/Byleen Business Calculus 12e Answer to Revenue Problem B) Revenue = Quantity  Price The domain of this function is the same as the domain of the price-demand function, which is [1, 25] (in thousands.)

25 Barnett/Ziegler/Byleen Business Calculus 12e 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 thousands, C(x) is in thousands of dollars A)Write a profit function for producing and selling x thousand notebook computers, and indicate the domain of this function. B)Does the company make a profit/loss if 20 thousand notebooks are made and sold?

26 Barnett/Ziegler/Byleen Business Calculus 12e Answer to Profit Problem A) 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 (x is in thousands of notebooks) B) P(20) = 2000 (in thousands of dollars) The profit is $2,000,000 when 20,000 notebooks are made and sold Thousand dollars Thousand notebooks

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