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Functions And Mathematical Functions And Mathematical Models Models – Your height is a function of your age. Your height is a function of your age. – The cost of mailing a package is a function of weight. The cost of mailing a package is a function of weight. – The area of a circle is a function of its radius. The area of a circle is a function of its radius. – The weight of an astronaut is a function of her elevation. The weight of an astronaut is a function of her elevation. – The price of a commodity is a function of the demand for that The price of a commodity is a function of the demand for that commodity. commodity. – Your exam grade is a function of the time you spend studying!!! Your exam grade is a function of the time you spend studying!!! A function is a “rule” that describes how one quantity A function is a “rule” that describes how one quantity depends on another. For example depends on another. For example One of the most basic and important ideas in all One of the most basic and important ideas in all mathematics is the concept of a function. mathematics is the concept of a function. Fencing Problem 1 Fencing Problem 1 A rancher has 200 feet of chain-link fencing to enclose a rectangular field as demonstrated below. How should the rancher cut the chain-link fencing in order to maximize the enclosed area? 200 feet l w Numerical Representation Numerical Representation Widt h Length Area 0 100 0 5 95 475 10 90 900 15 85 1275 20 80 1600 25 75 1875 30 70 2100 35 65 2275 40 60 2400 45 55 2475 50 50 2500 55 45 2475 60 40 2400 65 35 2275 70 30 2100 75 25 1875 80 20 1600 85 15 1275 90 10 900 95 5 475 100 0 0 105 -5 -525 Graphical Representation Graphical Representation ( ) ( ) 100A w w w= − “The value of the function A at w” OR “A of w” Dependent variable Independent variable Algebraic Representation Algebraic Representation ( ) ( ) 100A w w w= − What is the domain of the area function A ? OR What are the possible values for the width w? [ ] { } 0,100 | 0 100Domain w w = = ≤ ≤ [ ] 0 100 Domain and Range Domain and Range Domain Range Fencing Problem 2 Fencing Problem 2 A rancher has 200 feet of chain-link fencing to enclose two adjacent rectangular corrals as demonstrated below. How should the rancher cut the chain-link fencing in order to maximize the enclosed area? 200 feet l w