1. MAT 213 Brief Calculus Section 1.1 Models, Functions and Graphs

2. Mathematical Models The process of translating a real world problem situation into a usable mathematical equation is called mathematical modeling For example, in business it is important to know how many units to produce to maximize profit –Thus if we can model our profits as a function of the number of units produced, we can use calculus to determine how many products will maximize our profit

3. A function is a rule that assigns exactly one output to every input In mathematics, a function is often used to represent the dependence of one quantity upon another We therefore define the input as the independent variable, and the resulting output as the dependent variable Note: the output does not have to depend on the input in order to have a function

4. Definitions Independent Variable  Its values are the elements of the DOMAIN  Plotted on the horizontal axis  Its values are known when collecting data Dependent Variable  Its values are the elements of the RANGE  Plotted on the vertical axis  The quantity measured for a specific value of the independent variable.

5. Is it a function??? InputOutput -2 0 1 2 9753197531

6. Is it a function??? InputOutput 5555555555 -2 0 1 2

7. Is it a function??? InputOutput -2 0 1 2 5555555555

8. What about these? A = {(0,4), (7,4), (5,3), (1,0)} B = {(0,1), (1,1), (1,0)} C = {(1,1)}

9. Is it a function???

11. We use the notation f(x) to denote a function. It is read "f of x," meaning the value of the function f evaluated at point x Actually, any combination of letters can be used in function notation Example: If we were writing a function that described the area of a square in terms of the length of a side, we may choose A(s) to mean the area A when the side is length s. The parentheses DO NOT mean multiplication!!! Function Notation

12. Examples Find the function values. Do not worry about simplifying right now h(x) = x 2 + 2x - 4 a. h(4) b. h(-3x) c. h(a – 1) d. h(x+1) – 3h(x)

13. g(x)g(x) 1.g(-2) = ? 2.g(-1) = ? 3.Find the values of x that make g(x) = 0.

14. Rule of Four Functions can be represented in 4 ways 1.Numerical data such as a table 2.Graphically 3.In words 4.By an equation We will encounter all 4 of these representations during the semester

15. In Business Fixed costs (overhead) Variable costs Total Cost = Fixed costs + Variable costs Average cost = ----------------------------- Profit = Revenue – Total Cost When does a company break even?

16. Break-Even Point $ Revenue Total Cost 10 20 30 Number of Units (in millions) How many units would this business need to sell in order to break even?

17. Break-Even Point $ Profit 5 10 15 How many units would this business need to sell in order to break even? # of units (in thousands)

18. Combinations of Functions Now if we have a revenue function, R(x), and a cost function, C(x), we saw that we can create a profit function, P(x) We would get P(x) = R(x) - C(x) Thus we have combined two functions via subtraction to get another function We can also add, multiply or divide two functions

19. Composition of Functions Notation Take the functions f(x) and g(x) f(g(x)) = (f◦g)(x) To evaluate f(g(x)), always work from the inside out. First find g(x) then plug that result into f. For all x in the domain of f such that f(x) is in the domain of g

20. Composition of Functions Example Let f(x) = 5x + 1 and g(x) = x 2 Evaluate the following: (f◦g)(x) (f◦f)(x) g(f(x)) f(g(-2))

21. In groups let’s try the following from the book 1, 13, 25, 27, 35, 53