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GENERAL MATHEMATICS ppt.....( functions)
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- 2. Relation Function A relationis a rule that relates values 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 pair 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 such that no two ordered pairs have the same but different
- 3. Domain I L O V E M A T H Range 2 3 4 5 6 8 {( 𝑰, 4),(𝑳,5),(𝑶,6),(𝑽 ,8),(𝑬 ,3),( 𝑴,6),( 𝑨 ,2),(𝑻 ,8),(𝑯 , 4)} The above relation is function, since each element in the domain is assigned to one element in the range. Even though the range element 4 is assigned to two elements of the domain, letter H and I, it is still a function since each element of the domain is assigned to only one element of the range.
- 4. 7 1 2 Relation I 11 23 19 17 13 The mappingdiagram above is not a function. The relation I is not a function because there is at least on element in X for which there is more than one corresponding for example, corresponds to Similarly, corresponds to both
- 5. Target No. ofShirt Sales Price per T-shirt 500 900 1300 1700 2100 2500 ₱540 ₱460 ₱380 ₱300 ₱220 ₱140 The price for which you can sell x printed t-shirt is called the price function represents each data point in the table. Example: To sell more T-shirts, the class needs to charge lower price as indicated in the following table:
- 6. To writea linear equation with two variables when given two points using the slope-intercept method, you have to: find the slope of the line using the slope formula and - write the linear equation with two variables by substituting the values of m and to the formula the point-slope form of a linear equation. Using the points and we have:
- 7. Now, substituteone of the ordered pairs and the slope, X Thus, the price function is Note: To check, verify that the function fits each data in the table.
- 8. Example of functions Onehundred meters of fencing is available to enclose a rectangular area next to a river (see figure). Give a function A that can be enclosed, in terms of x. 𝑦 𝑥 river 𝑨=𝒙𝒚
- 9. Solution: Letx and y denoted the lengths of the sides of the garden. Then the area A Express A in terms of a single variable, either x or y. the total perimeter is 100 meters. Hence,
- 10. Piecewise functions A piecewise function or a compound function is a function defined by multiple subfunction applies to a certain interval of the main function’s domain. Some situations can only be described by more than one formula, depending on the value of the independent variable. Example: A user is charged 300 monthly for a particular mobile plan, which ₱ includes 100 free text messages. Messages in excess of 100 are charged 1 ₱ each. Represent the amount a consumer pays each month as a function of the number of messages sent in a month.
- 11. Solution: letrepresent the amount paid by the consumer each month. It can be expressed by the piece function
- 12. Evaluating function Evaluate function means replacing the variable in the function, in this case, with a value from the function’s domain and computing for the result. To denote that we are evaluating at for some in the domain of we write. The function notation tells you that y is a function of x. If there is a rule relating to x , such as then you can also write: 1
- 13. To find fora given value of x is to evaluate the function by substituting the input value in to the equation. The domain is the set of all that makes sense in the equation.
- 14. Example: If, evaluateeach. a. b. c. d. Solution: Substituting the value of
- 15. Another Example: If Find: a. b. Solution: To find, you let. Since -4 is less than 0, you use the first line of the function. Thus, To find, you let . Since 3 is greater than 0, you use the second line of the function. Thus,
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- 17. For example: Addthe terms on the right-hand side of the equal sign of to the terms on the right-hand side of the equal sign of. Function, like numbers, can be added, subtracted, multiplied, or divided. Because functions are usually given in equation form, you perform these operations by applying the algebraic expressions.
- 18. Sum, Difference, Product,and Quotient of Functions Let and be any two functions. The sum, difference, product, and quotient are functions whose domains are the set of all real numbers common to the domain of , and defined as follows: Sum: Difference: Product: Quotient: , where
- 19. Examples: Ifand, find: A) b) c) d) Determine the domain of each function. a. b. Definition of sum of function Add f(x) and g(x) Combine like terms Solution: (f + g)x = f(x )+ g(x) =3x-2+ +2x-3 = +5x-5 Definition of the difference of functions. Subtract Perform the subtraction Combine like terms
- 20. c.) d) The domain ofis the set of all real numbers except Definition of product of functions. Apply distributive property. Apply distributive property. Combine like terms.
- 21. The Composition ofFunctions The Composition of the function is denoted by and is defined by the equation: The domain of composition function is the set all x such that 1. X is in the domain of ; and 2. is in the domain of f .
- 22. Example: Givenand , find: Solution: Because means, you must replace each occurrence of in the function . f Thus, Give equation for . Replace Replace Apply Distribution Law. Combine like terms
- 23. Answer thefollowing functions. Write it in a one whole sheet of paper. Evaluate : (1 point each item) Given , find the following values: 1. 2. 3. Given , find the following values: 2. 3.
- 24. Perform thefollowing functions. (5 points each item) Given functions and below, find and simplify the following functions: 1. 2.
- 25. GOOD LUCK ! LOIDM.TERMO