Lesson 1 Review on Functions.pptxdsgnsfjngkjsrnkgmknkjnjkjnjnnj

This lesson is all about function.

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2. OPENING PRAYER OPENING PRAYER OPENING PRAYER Our Father, who art in heaven, hallowed be thy name; thy kingdom come; thy will be done on earth as it is in heaven. Give us this day our daily bread; and forgive us our trespasses as we forgive those who trespass against us; and lead us not into temptation, but deliver us from evil. Amen.

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4. OBJECTIVE/S: represents real-life situations using functions. evaluates a function.

5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Does the diagram below illustrate a function? Why or why not?

6. FUNCTION

7. What is function? A function is a relation defined as a set of ordered pairs (x, y) where no two or more distinct ordered pairs have the same first element (x); that is, every value of x corresponds to a unique value of y. A function can be named using any letter of the English alphabet.

8. Examples: Determine whether each of the following is a function or not. a. f = {(0, -1), (2, -5), (4, -9), (6, -13)} b. r = {(a, 0), (b, -1), (c, 0), (d, -1)} c. g = (5, -19), (25, -75), (50, -100) d. t = {(-2, 0), (-1, 1), (0, 1), (-2, 2)}

9. Solution: *The examples given in (a) and (b) are functions since no two ordered pairs have the same values of x. *Example (c) is not a function as it does not indicate a set. It is simply a listing of ordered pairs. *Example (d) is not a function because there are ordered pairs having the same first element x.

10. A table of values is commonly observed when describing a function. This shows the correspondence between a set of values of x and a set of values of y in tabular form. Examples: x 1 4 9 16 y -5 -4 -1 4 11 x -2 -1 1 2 y -2 1 3 -1 1

11. A function can also be expressed as a correspondence or mapping of two nonempty sets. Examples: *In figure 1, F is a function since each element in x corresponds to a unique element in y. *In figure 2, G is not a function. Notice that the element 2 in x corresponds to two values of y, which are -3 and 4.

12. The idea of a function can also be extended to real- life situations. For example, plants rely on the amount of sunlight and rainfall for growth. The area of a square is defined by the length of its side. In the same manner, the relationship that exists a chemical element and its corresponding chemical symbol also indicates a function.

13. Examples: Identify whether the relationship that exists between each of the following pairs indicates a function or not. a. A jeepney and its plate number. b. A teacher and his cellular phone. c. A student and his ID number. d. A pen and the color of the ink.

14. Solution: a. Function. A jeepney can only be assigned one plate number. b. Function. A student may only be issued one ID number. c. Not a function. A teacher may have two or more cellular phones. d. Not a function. There are some pens that have two or three colors of inks contained in only one unit.

15. Another way to determine functions is through their graphs. Recall, that in function, every value of the independent variable, say x, corresponds to a unique value of the dependent variable, y. Therefore, any vertical line drawn through the graph of a function must intersect the graph at exactly one point. This is known as the vertical line test for a function. Note: *The vertical line test for a function states that if each vertical line intersects a graph in the x- y plane at exactly one point, then the graph illustrates a function.

16. Examples : Which of the following graphs illustrates a function?

17. Solution: *The graph in (a) and (b) illustrate a function. Notice that when a vertical line is drawn over these graphs, the line will intersect each graph at exactly one point. *The graph in (c) is not a function because any vertical line will intersect that graphs at two or three points.

18. Solution: *The graph in (a) and (b) illustrate a function. Notice that when a vertical line is drawn over these graphs, the line will intersect each graph at exactly one point. *The graph in (c) is not a function because any vertical line will intersect that graphs at two or three points.

19. The relationship between the variables x and y can be denoted by the equation y = f(x). This rule allows you to determine the unique value of y for every given value of x. The variable x is the independent variable, while the variable y is the dependent variable. For example, the area A of a square is determined by the equation A = , and the circumference C of a circle is given the equation C = 2 r. This means that the length of the side s and a square determines its area, and the length of the radius r of a circle determines its circumference.  

20. Since y = f(x), y = 2x – 1, for instance can also be written as f(x) = 2x – 1. The relationship between the independent and dependent variables, defined by an equation or a rule, is summarized in figure 3.

21. Examples: 1. Find the value of y in the equation y = 3x – 2 if x = -1. Solution: Substitute the value of x into the given equation, then solve for y. y = 3x – 2 = 3(-1) – 2 = -5 *Therefore, in the given equation, if x = -1, then y = -5. These values can be written as an ordered pair (-1, -5).

22. 2. If the value of y in the equation y = is 2, find x. Solution: Replace y with 2 in the equation, then solve for x. 2 = 2(x – 2) = 3x + 8 Multiplying both sides by x - 2 2x – 4 = 3x + 8 Distributing property of multiplication 2x – 3x = 8 + 4 Combining similar terms -x = 12 x = -12 Dividing both sides by -1 *Therefore, x = -12 if y = 2 in the given equation.  

23. 3. The volume of a cube is defined by the function V = , where e is the length of an edge. What must be the length of an edge of a cube if its volume is 2 197 ? Solution: Substitute the volume of the cube into the formula and extract the cube root of both sides of the equation. V = 2 197 = = e = 13 mm  

24. 3. The volume of a cube is defined by the function V = , where e is the length of an edge. What must be the length of an edge of a cube if its volume is 2 197 ? Solution: Substitute the volume of the cube into the formula and extract the cube root of both sides of the equation. V = 2 197 = = e = 13 mm  

25. 4. A car has traveled a distance of 42 kilometers (km) in 3 hours (h). Find the speed of the car. Solution: Speed is determined by the formula s = , where d is the distance traveled in a given time t. Therefore, s = = = 14 km/h. *The speed of the car is 14 km/h.  

26. Evaluating Functions You have learned in the previous discussion that a function can be presented in a form of an equation. The number assigned to a given variable determines the value of the function at that number. This process is known as evaluating function . When you evaluate a function, it means that you are going to solve for the function value given a particular value of the variable used in the equation.

27. Examples: 1. Let f be a function defined by f(x) = 5x -3. Find the following: a. f(-2) b. f( ) c. f(3) + f(-3) d.  

28. Solution: Replace the variable x in the equation by the given value of x in each item. Then simplify. a. f (-2) = 5(-2) – 3 = -13 b. f( ) = 5( ) – 3 = – 3 =  

29. THANK YOU! FOR LISTENING…