1. Copyright © Cengage Learning. All rights reserved. 4 4 Functions

2. Copyright © Cengage Learning. All rights reserved. Velocity P.1 Section Functions 4-1

3. 3 Functions Many quantities in everyday life are related to one another or depend on one another. A fundamental idea in mathematics and its applications is that of a function, which tells how one thing depends on another. When the value of one quantity uniquely determines the value of the second quantity, we say the second quantity is a function of the first. One way to express these relationships is using ordered pairs (x, y). The x variable is called the independent variable and the y variable is called the dependent variable, because its value is dependent on the value of x.

4. 4 Functions Generally, a function can be represented in four ways: Description or rule Table of values Graph Equation or formula In mathematics, a relation associates members of one set with members of another set. A function is a special kind of relationship between two quantities.

5. 5 Functions Specifically, in a function it must be true that for each value of the independent variable, there is exactly one value for the dependent variable.

6. 6 Example 1 – The Cricket Function The snowy tree cricket is called the thermometer cricket because an accurate estimate of the current temperature can be made using its chirp rate. One can estimate the temperature in degrees Fahrenheit by counting the number of times a snowy tree cricket chirps in 15 seconds and adding 40. Use R as the chirp rate per minute and describe this function with words, a table, a graph, and a formula. Words: Because R represents chirps per minute, we will count the value for R and divide it by 4 to get the number of chirps in 15 seconds. Then add 40 to that number to estimate the temperature.

7. 7 Example 1 – The Cricket Function Table: See Table 4-2. Notice that each 20 chirps per minute increase corresponds to a 5  increase in temperature. cont’d

8. 8 Example 1 – The Cricket Function Graph: The values in the table are plotted on the graph in Figure 4-2. The graph illustrates that if more chirps per minute are counted, the corresponding temperature is warmer. cont’d Figure 4-2 Example 1.

9. 9 Example 1 – The Cricket Function Formula: An algebraic rule can be developed using the description as follows: We can use this formula to calculate the estimated temperature for any number of chirps per minute. cont’d

10. 10 Functions The key concept in a functional relationship is that there is only one value of the dependent variable (y) for each value of the independent variable (x). Think of the independent variable as the input of the function. This is the variable that you control—the one that you choose or determine. Once you have chosen the value of the independent variable, there will be only one dependent variable or output that corresponds to it. (See Figure 4-5.) Figure 4-5

11. 11 Example 3 – Determining Functional Relationships Determine the independent and dependent variables in the following functions. Give a statement that illustrates the functional relationship described. (a) The altitude and temperature when climbing a mountain. The altitude is the independent variable (input) and the temperature at that altitude is the dependent variable (output).

12. 12 Example 3 – Determining Functional Relationships As the altitude of the climb increases, the surrounding temperature will decrease. This is illustrated in Figure 4-6. Figure 4-6 cont’d

13. 13 Example 3 – Determining Functional Relationships (b) The speed of a car and its stopping distance The stopping distance of a car is dependent on the speed of the car when the brakes are applied. (This is the reason that policemen measure the length of the skid marks at the scene of an accident.) Therefore, the speed of the car is the independent variable (the one that you choose) and the braking distance is the dependent variable (a result of the speed that the car was travelling). cont’d

14. 14 Example 3 – Determining Functional Relationships The faster the car goes, the longer the stopping distance will be. Table 4-3 gives some stopping distances for cars traveling at different speeds on dry pavement. cont’d

15. 15 Example 3 – Determining Functional Relationships (c) The speed of a baseball dropped from the top of a tall building as shown in Table 4-4. cont’d

16. 16 Example 3 – Determining Functional Relationships If a baseball is dropped from the top of a tall building, the speed of the baseball is uniquely determined by the amount of time that it has been falling. Therefore, its velocity is a function of the time that has passed since it was dropped. As the length of time since it was dropped increases, the baseball’s velocity continues to increase. cont’d

17. 17 Example 3 – Determining Functional Relationships (d) The formula for calculating the area of a circle: A =  r 2 The area of a circle (A) is dependent on the radius of the circle (r) (see Figure 4-7). cont’d Figure 4-7

18. 18 Example 3 – Determining Functional Relationships As the radius of the circle changes, a corresponding change in the area will occur. So the radius is the independent variable (input), and the area is the dependent variable (output) in this function. cont’d