General Mathematics Function Leson 1-WEEK2.pptx

its about function

1. GENERAL MATHEMATICS LESSON 1: FUNCTION

2. FUNCTIONS LEARNING OUTCOMES : at the end of the lesson, the learner is able to represent real-life situations using functions, including piece-wise functions. LESSON OUTLINE Functions and relations Vertical line test Representing real-life situations using functions, including piece wise functions.

3. FUNCTION MACHINES

4. Functions and relations Definition: a relation is a rule that relates values from a set of values (called the domain ) to a second set of values (called the range ) A relation is asset of ordered pairs ( x,y ) Definition: a function is a relation where each element in the domain is related to only one value in the range by some rule.

5. Functions and relations 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. Using functional notation, we can write f(x) = y, read as “ f of x is equal to y”. In particular, if (1,2) is an ordered pair associated with the function f, then we say that f(2)=1 Example1. which of the following relations are functions? f= (1,2), (2,3), (3,5), (4,7) g= (1,3), (1,4), (2,5), (2,6), (3,7) h= (1,3), (2,6), (3,9), …., (n,3n)

6. Solution The relations f and h are functions because no two ordered pairs have the same x-value but different y-values. Meanwhile, g is not a function because (1,3) and (1,4) are ordered pairs with the same x-value but different y-values.

7. Relations and functions can be represented by mapping diagrams where the elements of the domain are mapped to the elements of the range using arrows. In this case, the relation or function is represented by the set of all the connections represented by the arrows.

8. Solution The relations f and g are functions because each value y in Y is unique for a specific value of x. the relation h is not a function because there is atleast one element X for which there is more than one corresponding y-value. For example x=7 corresponds to y=11 or 13. Similarly, x=2 corresponds to both y=17 or 19.

9. Vertical line test A relation between two sets of numbers can be illustrated by a graph in the cartesian plane, and that a function passes the vertical line test. The vertical line test A graph represents a function if and only if each vertical line intersects the graph at most once.

10. Vertical line test Solution Graphs a), b),c) are graphs of functions while d) and e) are not because they do not pass the vertical line test.

11. VERTICAL LINE TEST 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 tat relate only one value from the set of outputs to a value from the sets of inputs.

12. Domain and range of functions Definition: The domain of a relation is the set of all possible values that the value x can take . The range of a relation is the set of all possible values that the variable y can have.

13. Domain and range of functions Example 4: Identify the domain for each relation using set builder notation. (a) y=2x+1 (b) y= (c) y= x+1   S  

14. Representing real-life situations using functions including, Functions can often be used to model real situations. Identifying an appropriate functional model will lead to a better understanding of various phenomena Example 5. Give a function C that can represent the cost of buying x meals, if one meal costs P40 Solution. Since each meal costs P40, then the cost function is C(x)=40x.

15. Representing real-life situations using functions including, Example 6. 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. Solution. The area of rectangular enclosure is A= xy . We write this as function of x. since only 100m of fencing is available, then x+2y=100 or y=(100-x)/2=50-0.5x. Thus, A(x)= x(50-0.5x)=50x-0.5  

16. Piece wise functions Some situations can only be described by more than one formula, depending on the value of the independent variable. Example 7 . 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. Solution. The cost of text messaging can be expressed by piece wise function.

17. Representing real-life situations using functions including functions Example 8. 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 a jeepney fare in terms of the distance (d) in kilometers. Solution. The input value is distance and the output is the cost of the jeepney fare. If F(d) represents the fare as a function of distance, the function can be represented as follows:

18. ITS YOUR TURN! 1. The relation (0,0), (1,1), (2,4), (3,9),….(n,n²) a function? 2. Can the graph of a circle be considered a function? 3. Which of the following letters will pass the vertical line test? V W X Y Z.

19. THANK YOU!