General Mathematics Slides - Function.pptx

Function

1. FUNCTIONS AND RELA TIONS Chapter 1

2. FUNCTIONS AS MO DELS At the end of this lesson, you should be able to represent real life situations using functions, including piecewise functions, and solve problems involving functions.

3. Lesson Outline ■ Review: R elations and Functions ■ Review: The function as a machine ■ Review: Functions and relations as a table of values ■ Review: Functions as a graph in the Cartesian plane ■ Review: Vertical Line Test ■ Functions as representations of real - life situations ■ Piecewise Functions

4. Functions and Relations Relations A relation is a rule that relates from a set of values (called the domain) to a second set of values (called the range) The elements of the domain can be imagined as input to a machine that applies a rule to these inputs to generate one or more outputs. A relation is also a set of ordered pairs (x, y) Functions A function is a relation where each element in the domain is related to only one value in the range by some rule. The elements of the domain can be imagined as input to a machine that applies a rule so that each input corresponds to only one output. A function is a set of ordered pairs ( x,y ) such that no two ordered pairs have the same x-value but different y-values.

5. Function as a machine

6. Functions and relations as a table of va lues

7. Functions and relations as a table of va lues one to one many to one one to many

8. Functions as a graph in the Cartesian p lane

9. Vertical Line Test Example 3. Which of the following graphs can be graphs of funct ions?

10. Vertical Line Test Example 3. Which of the following graphs can be graphs of funct ions?

11. Chec kpoint: ■ When is a relation said to be a fu nction? ■ Explain the vertical line test

12. Group Activity: 5 minutes ■ Answer the given worksheet in 5 minutes 5 4 3 2 1

13. Which of the following represents a fun ction? a) y=2x+1 b) y= x 2 -2x+2 c) x 2 + y 2 =1 d) y= + 1 e) y= 2 +1 − 1

14. The domain of a relation is the set of all possible values that the variable x can take a) y=2x+1 b) y= x 2 -2x+2 c) x 2 + y 2 =1 d) y= + 1 e) y= 2 +1 − 1

15. If a relation is a function, then y can be replaced with f(x) to denote that the value of y depends on the value of x. Replace y in the following examples to denote a fun ction: a) y=2x+1 b) y= x 2 -2x+2 c) y = + 1 d) y= 2 +1 − 1

16. If a relation is a function, then y can be replaced with f(x) to denote that the value of y depends on the value of x. Replace y in the following examples to denote a fun ction: a) f(x)=2x+1 b) q(x)= x 2 -2x+2 c) g(x)= + 1 d) r(x)= 2 +1 − 1

17. Chec kpoint: ■ Define function ■ When is a relation said to be a function? ■ What is the use of the vertical line test?

18. Important concepts ■ Relations are rules that relate two values, one from a set of inputs and the second from the set of outputs. ■ Functions are rules that relate only one value from the set of outputs to a value from the set of inputs.

19. Quiz: Determine whether each of the following relation is a function or not 1. {(1,-2), (-2,0), (-1,2), (1,3)} 2. {(1,1), (2,2), (3,5), (4,10), (5,15)} 3. y 2 = 3x+2 4. y=4-5x 5. 6. 7. 8.

20. Quiz: Determine whether each of the following relation is a function or not 9. 10.

21. FUNCTIONS AS REPRESENTATIONS OF REAL - LIFE SITUATIONS What are piecewise functions?

22. Using functions in real-life situations

23. One hundred meters of fencing is available to enclose a rectangular area next to a river. Give a function A that can represent the area that can be enclosed, in terms of x.

24. P IECEWISE FU NCTIONS Some situations can only be described by more than one formula

25. A user is charged P300 monthly for a particular mobile plan, which includes 100 free text messages. Messages in excess of 100 are charged P1 each. Represent the monthly cost for text messaging using the function t(m), where m is the number of messages sent in a month.

26. A jeepney ride costs P8.00 for the first 4 kilometers, and each additional integer kilometer adds P1.50 to the fare. Use a piecewise function to represent the jeepney fare in terms of the distance (d) in kilometers.

27. Water can exist in three states: solid ice, liquid water, and gaseous water vapor. As ice is heated, its temperature rises until it hits the melting point of 0°C and stays constant until the ice melts. The temperature then rises until it hits the boiling point of 100°C and stays constant until the water evaporates. When the water is in a gaseous state, its temperature can rise above 100°C (This is why steam can cause third degree burns!). A solid block of ice is at -25°C and heat is added until it completely turns into water vapor. Sketch the graph of the function representing the temperature of water as a function of the amount of heat added in Joules given the following information: The ice reaches 0°C after applying 940 J. The ice completely melts into liquid water after applying a total of 6,950 J. The water starts to boil (100°C) after a total of 14,470 J. The water completely evaporates into steam after a total of 55,260 J. Assume that rising temperature is linear. Explain why this is a piecewise fun ction.

28. Solution. Let T(x) represent the temperature of the water in degrees Celsius as a function of cumulative heat added in Joules. The function T(x) can be graphed as f ollows:

29. EVALUAT ING FU NCTIONS At the end of this lesson, you will be able to evaluate functions

30. Evaluating a function means replacing the variable in the function.

31. Solution: Substitute 1.5 for x

32. Se atwork