1. The length of an object in inches, as a function of its length in feet. 2. The length of an object in feet, as a function of its length in inches

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1 Functions Algebra is about relations. If we know how two quantities are related, then information about one quantity gives us information about the other. For instance, if we know the relationship between the length of the side of a square and the area of the square, then knowing one of these quantities is enough to determine the other. This is very important in math and science, because we can often use information about something that is easy to measure in order to learn something about something that is either difficult or impossible to measure. (Think about measuring the height of a mountain directly with a ruler - this is impossible. However, with trigonometry we can measure the height indirectly.) A function is a relation in which each input has at most one output. This is required for the relationship to behave as we would expect in the world. For instance, if we had a function representing how much revenue a company earned during each year, there can only be one value for each year. Functions will serve as the basis of our studies in algebra. To begin, let s consider some examples of relations we can represent with functions. Note that some of these functions may not represent the exact relationship, but rather serve as a simplified mathematical model of the situation. The purpose of a model is to approximate a real-world phenomenon. In terms of expressions and equations, we can think of each functions as being given by an expression. When we want to find determine certain values of a function, then we need to solve an equation. One side of the equation will be the expression for the function, and the other side will be the desired function value. As we will see, there are a number of different types of relations we will encounter. Some of these types are: (1) unit conversions, (2) geometric relations, (3) relationships between distance, speed, and time, (4) scientific models of the world, (5) relationships in economics, and (6) mathematical models of observed data. Now let s look at 20 example functions, to get a better idea of this. 1. The length of an object in inches, as a function of its length in feet 2. The length of an object in feet, as a function of its length in inches 3. The weight of an object on Earth in pounds, as a function of of its mass in kilograms 4. Temperature in degrees Fahrenheit, as a function of temperature in degrees Celsius 5. Area of a square, as a function of side length 6. Circumference of a circle, as a function of radius 7. Perimeter of a rectangle, as a function of the length and width of its sides 8. Perimeter of a rectangle with one side of length 5, as a function of the other side length 9. Distance traveled by a velociraptor running at a constant speed, as a function of time 10. Position of a person walking at a constant speed, as a function of time 11. Distance traveled by a brick dropped off of the roof of a building, as a function of time 12. Height of a brick dropped off of the roof of a building, as a function of time 13. Time required to travel a distance of 6 miles at constant speed, as a function of speed 14. Linear expansion of rubber, as a function of temperature in Celsius 6

2 15. Maximum heart rate during exercise, as a function of age 16. Target heart rate during exercise to improve cardiovascular conditioning, as a function of age 17. Cost of a purchase after sales tax, as a function of original cost 18. Value of a $1000 investment with an interest rate of 5%, as a function of years invested 19. Google s advertising revenue, as a function of year 20. The average number of meals per person Americans ordered for take out from 1984 to 2005, as a function of number of years after 1984 Now let us consider each of these functions in greater detail. 1. The length of an object in inches, as a function of its length in feet. Let I represent the length of the object in inches, and f the length of the object in feet. Since there are 12 inches in every foot, we find that I(f) = 12 f. This relationship tells us that someone who is 5 feet tall is 60 inches tall. What if we know someone is 48 inches tall. How many feet tall are they? To figure it out we set up an equation, 48 = 12 f. This equation tells us that we need to find a value for f that when multiply it by 12, we get 48. We can also think about creating a different function to represent length in feet in terms of length in inches. 2. The length of an object in feet, as a function of its length in inches. Once again, let I represent the length of the object in inches, and f the length of the object in feet. In this case there are 1/12 feet in each inch, so we have f(i) = 1 12 I. If we have a 72 inch pole, how many feet long is it? 3. The weight of an object on Earth in pounds, as a function of of its mass in kilograms Let W represent the weight of an object on Earth in pounds, and m its mass in kilograms. As a result of Earth s gravity, objects on Earth have a weight (in pounds) for 2.2 times their mass in kilograms. Thus, we find W (m) = 2.2m. As a quick way to estimate someone s weight given their mass, just multiply by 2. 7

3 4. Temperature in degrees Fahrenheit, as a function of temperature in degrees Celsius. Let F represent the temperature in degrees Fahrenheit, and C the temperature in degrees Celsius. F = 9 5 C If we look at the freezing and boiling points of water in degrees Celsius (0 and 100), we can see the corresponding values in degrees Fahrenheit are 32 and Area of a square, as a function of side length. Let A represent the area of the square, and x the length of a side. We find that A(x) = x x. What is the length of the side of a square if the area is 4? To find this we have to solve the equation What number times itself equals 4? 4 = x x. 6. Circumference of a circle, as a function of radius. A very important discovery is that the ratio between the circumference and diameter (twice the radius) of a circle is fixed. We use the symbol π to denote the ratio of this relationship. If we let C represent circumference and r radius, we find that C = 2πr. What if we want to find the radius of a circle with circumference 12? Then we need to solve the equation 12 = 2πr. How can we find the solution? And when we do, it is expressed in terms of π. If we want to use this length for a calculation, then we need to make an approximation for π, such as using Perimeter of a rectangle, as a function of the length and width of its sides. The perimeter of a rectangle depends on the length and width of its sides. Thus, this is a function that depends on two variables. If we let P represent perimeter, l the length of two sides, and w the length of the other two sides, we find that P (l, w) = 2l + 2w. In order to find the area we need two pieces of information, unlike in the previous formulas. However, if we fix one of the sides to a specific length, then we could have a function with only one input variable, as in the next example. 8

4 8. Perimeter of a rectangle with one side of length 5, as a function of the other side length. Let P represent the perimeter of the rectangle, and l the length of the undetermined side. Then we find, P (l) = l. What is the length of a side to get a perimeter of 8? Does it make sense to ask this question? 9. Distance traveled by a velociraptor running at a constant speed, as a function of time. Estimates are that a velociraptor could run up to about 40 miles per hour. If we let d represent distance traveled, and t represent time, then we have d(t) = 40t. How long does it take the velociraptor to travel 60 miles? 10. Position of a person walking at a constant speed, as a function of time. Now let s suppose we have a person (Dennis Nedry) jogging through the jungle, 5 miles away from the velociraptor, at a speed of 5mph. Let p represent Dennis position with respect to the the velociraptor, and t represent time. The first thing that we need to do is set up a coordinate system, with the raptor at 0, and Nedry at 5. Now we find the following equation, p(t) = 5 + 5t. If Nedry was standing still, how long would it take the velociraptor to catch him? We need to solve the equation 40t = 5. Thus, we find that it would take the velociraptor 1/8 of an hour, or hours. If we wanted this in minutes, we would solve 1 8 hours 60 minutes 1 hour = 60 8 minutes = 7.5 minutes. What if Nedry was running away (at the jogging speed)? Then we would have to solve 40t = 5 + 5t 35t = 5 t = 5 35 t = 1 7. If we convert to minutes, we find that 1/7 hours is about 8.6 minutes, so jogging away helps Nedry escape the raptor for about an additional minute. 9

5 11. Distance traveled by a brick dropped off of the roof of a building, as a function of time. Newton made the amazing discovery that if we ignore air resistance, all objects fall at the same rate near the surface of the earth. If we let the distance traveled by the brick be represented by d (in meters) and time (in seconds) by t, we find d(t) = 4.9t 2. If an object was dropped from a really high height, how far would it fall in 10 seconds? If we perform the calculation, we find it would fall 490 meters, which is about 1607 feet. If we consider a 10 story building is about 100ft, this is really far to fall in 10 seconds! 12. Height of a brick dropped off of the roof of a building, as a function of time. Now what if rather than calculating the distance traveled, we wanted to find the height of the brick we dropped, if we dropped it off of a 10 story building. Let h represent the height of the brick, t represent time (same units as before). We find, h(t) = t 2. Does this equation make sense for large values of t? Why or why not? 13. Time required to travel a distance of 6 miles at constant speed, as a function of speed. We found before that the distance d someone travels is the product of time traveled t and the speed s at which that person is traveling (if they are traveling at a constant speed). This gives us the relationship d(t) = r t. To find the time traveled as a function of distance and rate, we need to think about solving this equation for t. That means we need to find a number that when multiplied by the speed at which we are traveling, it gives us the distance traveled. But that number is just the distance traveled divided by the speed of travel, so we find t(d, r) = d r. Now if we know we have to travel a distance of 6 miles, then we find t(r) = 6 r. How long will it take us to travel 6 miles if we are going at 6 miles per hour? What about at 36 miles per hour? 14. Linear expansion of rubber, as a function of temperature in Celsius. When materials are subjected to changes in temperature, they either expand or contract as a result. For a 1cm length of rubber, this relationship can be modeled in one dimension as follows, L(T ) = T

6 where L is given in meters. This relationship assumes the starting temperature is 20C (because materials expand/contract at slightly different rates depending on their temperature). All temperatures expand and contract at different rates, which can be a problem when an object is made of many materials. For instance, different components of a bowling ball are different materials, so if it is subjected to too cold of temperatures, it can crack. 15. Maximum heart rate during exercise, as a function of age. If we let H represent the maximum heart rate, and a the person s age, then a simplified model is given by H(a) = 220 a. What happens to maximum heart rate as age goes up? 16. Target heart rate during exercise to improve cardiovascular conditioning, as a function of age. If we let H represent the target heart rate, and a the person s age, then a simplified model is given by H(a) = 4 (220 a). 5 What is the relationship between maximum and target heart rate? 17. Cost of a purchase after sales tax, as a function of original cost. In San Francisco, the sales tax is 8.75%. If we let T represent the cost of an item after taxes, and C the cost before taxes, we find that T (C) = C. How much tax does the city collect on a purchase of $200? 18. Value of a $1000 investment with an interest rate of 5%, as a function of years invested. Let V represent the value of the investment, and t the time, in number of years. After a year of investment, the original value will have increased by 5%. Another way to think of this is that after a year we will have 105% of the original investment. After another year, we will have 105% of the value we had at the beginning of the second year, or t = t. This pattern continues, and we find the relationship that tor an investment of this type, the value is given by V (t) = 1.05 t Google s advertising revenue, as a function of year. When we have real-world data, it s often not possible to find a symbolic formula to represent the functional relationship between two quantities. In these situations, a table can be a very useful tool. 11

7 Advertising Revenue (billions of USD) Year To put this in perspective, the budget for the entire California Department of Education was only $ billion! ( 20. The average number of meals per person Americans ordered for takeout from 1984 to 2005, as a function of number of years after Using data collected after the year 1984, a model was constructed of how many meals Americans ordered from takeout. If we let T represent the number of meals ordered per person, and x the number of years after 1984, we find that T (x) = 0.125x x How many takeout meals were ordered by the average American in 1987? 12