1. 1 MATH 1101 Introduction to Mathematical Modeling

2. 2 Section 1.1 Functions and Mathematical Models

3. 3 EXAMPLE 1 Number of EggsPrice of Breakfast 1$1.70 2$2.15 3$2.50 4$2.85 Price of Breakfast at Waffle House from Menu © 2002

4. 4 EXAMPLE 2 from: http://www.j-bradford-delong.net/TotW/g7.html Median Family Income

5. 5 EXAMPLE 3 s s Area of a Square A = s 2

6. 6 WHAT DO THESE EXAMPLES HAVE IN COMMON? They are all functions. Definition: A function f defined on a collection D of numbers is a rule that assigns to each number x in D a specific number f (x).

7. 7 DOMAIN AND RANGE The collection (or set) D of all numbers for which the number f (x) is defined is called the domain (of definition) of the function f. The set of all possible values y = f (x) is called the range of the function f.

8. 8 REMARKS ON FUNCTIONS The symbol f ( ) is regarded as an operation that is to be performed. The variable x is called the independent variable (or input). The variable y is called the dependent variable – because the value of y depends on x. The variable y is also called the output.

9. 9 MATHEMATICAL MODELING The process of describing (or modeling) a real- world situation with a function (or other mathematical equations) is called mathematical modeling. The resulting function (or equations) is called a mathematical model.

10. 10 A MODELING PROBLEM Suppose you own a fencing business. Your customer wants a rectangular pen for her golden retriever. She wants the pen to have area of 500 square feet. After discussing her budget, she can only afford 70 feet of fencing. However, the back wall of her house can be used as one of the four sides of the pen. What should you make the dimensions of the pen so that the area will be 500 ft 2 and the three sides total 70 ft?