1_Representation_ggggggggggggggggggggggggof_Functions.pptx

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1. REPRESENTATION OF FUNCTIONS

2. RELATION A relation is a set of ordered pairs. The domain of a relation is the set of first coordinates. The range is the set of second coordinates. Example of Relations {(1, 4), (2, 5), (3, 6), (4, 8)} {(4, 2), (4, -2), (9, 3), (9,3)} {(1, a), (1, b), (1, c), (1,d)}

3. FUNCTIONS A function is a relation in which each element of the domain corresponds to exactly one element of the range. Examples of Functions {(1, 4), (2, 5), (3, 6), (4, 8)} {(2, 1), (3, 1), (4, 1), (5,1)}

4. REVIEW Determine if the following relations represent a function. {(q, 0), (w, 1), (e, 2), (t, 3)} {(-1, -2), (0, -2), (1, -2), (2, -2)} {(1, 0), (1,1), (1, 2), (1, -2)} {(x, 3), (y, 4), (z, 3), (w, 4)} FUNCTION FUNCTION NOT FUNCTION FUNCTION

5. SOME TYPES OF FUNCTIONS LINEAR FUNCTION A function f is a linear function if f(x) = mx + b, where m and b are real numbers, and m and f(x) are not both equal to zero. QUADRATIC FUNCTION A quadratic function is any equation of the form f(x) = ax 2 + bx + c where a , b , and c are real numbers and a 0.  

6. SOME TYPES OF FUNCTIONS CONSTANT FUNCTION A linear function f is a constant function if f(x) = mx + b, where m = 0 and b is any real number. Thus, f(x) = b. IDENTITY FUNCTION A linear function f is an identity function if f(x) = mx + b, where m = 1 and b = 0. Thus, f(x) = x.

7. SOME TYPES OF FUNCTIONS ABSOLUTE VALUE FUNCTION The function f is an absolute value function if for all real numbers x, f(x) = x, for x ≥ 0 –x, for x ≤ 0 PIECEWISE FUNCTION A piecewise function or a compound function is a function defined by multiple sub-functions, where each sub-function applies to a certain interval of the main function's domain.

8. EXAMPLE Sketch the graph of the given piecewise function. What is f(– 4)? What is f(2)? x + 2, if x ≥ 0 –x 2 + 2, if x < 0 f(x)=

9. EXERCISE A Determine whether or not each relation is a function. Give the domain and range of each relation. {(2, 3), (4, 5), (6, 6)} {(5, 1), (5, 2), (5, 3)} {(6, 7), (6, 8), (7, 7), (7, 8)} FUNCTION NOT FUNCTION NOT FUNCTION

10. EXERCISE B Tell whether the function described in each of the following is a linear function, a constant function, an identity function, an absolute value function, or a piecewise function. f(x) = 3x − 7 g(x) = 12 f(x) = 3, if x > −5 -6, if x < −5 LINEAR FUNCTION CONSTANT FUNCTION PIECEWISE FUNCTION

11. EXERCISE B Tell whether the function described in each of the following is a linear function, a constant function, an identity function, an absolute value function, or a piecewise function. 4. 5. LINEAR FUNCTION ABSOLUTE VALUE FUNCTION

12. EXERCISE C A Zumba instructor charges according to the number of participants. If there are 15 participants or below, the instructor charges ₱500.00 for each participant per month. If the number of participants is between 15 and 30, he charges ₱400.00 for each participant per month. If there are 30 participants or more, he charges ₱350.00 for each participant per month. Write the piecewise function that describes what the instructor charges.

13. SOLUTION  

14. EXAMPLE 2 To sell more T-shirts, the class needs to charge a lower price as indicated in the following table: The price for which you can sell x printed T-shirts is called the price function p(x). p(x) represents each data point in the table.

15. Solution to Example 2 Step 1: Find the slope m of the line using the slope formula m = y 2 – y 1 / x 2 – x 1 Step 2: Write the linear equation with two variables by substituting the values of m and (x 1 , y 1 ) to the formula y – y 1 = m(x – x 1 )—the point-slope form of a linear equation. y – y 1 = m(x – x 1 ) y – 540 = −15 (x − 500) y – 540 = − 15 x + 100 y = − 15 x + 640 y = 640 – 0.2x Thus, the price function is p(x) = 640 – 0.2x.

16. Example 3 Find the dimensions of the largest rectangular garden that can be enclosed by 60 m of fencing.

17. Solution to Example 3 Let x and y denote the lengths of the sides of the garden. Then the area A = xy must be given its maximum value. Express A in terms of a single variable, either x or y. The total perimeter is 60 meters. 2x + 2y = 60 x + y = 30 y = 30 – x Hence , A = xy A = x(30 – x) A= 30x – x 2

18. Solution to Example 3 Write this equation in the vertex form by completing the square. A = –(x 2 – 30x + 225) + 225 A = –(x – 15) 2 + 225 The maximum area is 225 square meters. Since x = 15 (the width) and 30 – x = 15 (the length), the dimension that gives the maximum area is 15 meters by 15 meters.

19. Solution to Example 4 To the right of the y-axis, the graph is a line that has a slope of 1 and y-intercept of 2. To the left of the y-axis, the graph of the function is a parabola that opens downward and whose vertex is (0, 2). To sketch the graph of the function, you can lightly draw both graphs. Then darken the portion of the graph that represents the function.

20. Solution to Example 4 To find the value of the function when x = – 4, use the second equation. f(– 4) = – (– 4) 2 + 2 = – 16 + 2 = – 14 To find the value of the function when x = 2, use the first equation. f(2) = 2 + 2 = 4