General Mathematics - Evaluating Functions.pptx

Evaluating a function means substituting a specific value for the variable (like 'x') in the function's formula and then simplifying the resulting expression to find the output value

1. Evaluating Functions General Mathematics

2. For you to evaluate functions you have to first know the different parts of an expression.   An expression may contain the output variable and an equation containing one or more terms which consists a coefficient (consists of a numerical value) and / or a variable . Expression Terms Coefficient Variable

3. Monomial Binomial Trinomial Polynomial   An expression which contains only one term is called a monomial , two terms will be a binomial , three will be a trinomial , and expressions with more than three terms are called as polynomial .

4.   A term may or may not have an exponent attached on its variable. This exponent determines the power of the term whereas the higher the exponent, the higher the power of the term is. Those terms with higher power commonly placed first in the expression. or       → Descending Power →

5. Evaluating Functions In evaluating expressions or functions, you have first to simplify them by arranging the terms according to their power. ↓   The operator beside the term must be with the term on its left when arranging them.

6. Then perform the operation between like terms or those with the same power . ↓   If the coefficient becomes 1 , then the coefficient must be removed. Examples → → → → Multiplication and Division are the only operation that can modify the power of the term.  

7. Get the value of then input it into the equation . Then further evaluate the equation until you reach the output .   let ↓ ↓ ↓ ↓   Use the rule PEMDAS on the proper sequence of operation on evaluating equations

8. Another Example let ↓ ↓ ↓ ↓  

9. Another Example let ↓ ↓   let ↓ ↓ ↓  

10. Evaluating Function with Series of Inputs If you are given series of inputs, just evaluate the function with each input given then list down the result through either a set of ordered pairs, tables of values, or in a graph. Let With the given input values, you have to repeat the computation 4 times; each repetition uses one of the given value on the problem.  

11. ↓   ↓   ↓   ↓  

12. Collect your results and present them into these examples. Set of Ordered Pairs   Table of Values

13. In a graph, just plot the coordinates of each pair onto the cartesian plane and connect them all with a line  

14. ↓   Another Example: Where   ↓   ↓  

15.   You may extend the line on the graph as long as it correlates to the type of function your have.

16. Converting Word Problems into Equations First you have to identify what is ask. Ussually it is found after the words “How” or “What” and the second word will determine what value should we look for . Example: You want to buy apples on a local market, each apple they sell costs 15 pesos. How much it will cost you to buy 10 apples? On this problem, what is required is the total cost of the amount of apples you intended to buy and this value should be related to money.

17. Example: You want to buy apples on a local market, each apple they sell costs 15 pesos . How much it will cost you to buy 10 apples ? Then look for the coefficient which is something that cannot be changed on that situation such as the price of goods. While the input value must be the one that can be changed in that situation such as the number of goods you have to buy. The problem shows the coefficient is the price of the goods (15 pesos) , since it cannot be changed on that situation. While the input value is the number of goods that you have to buy (10 apples) .

18. Example: You want to buy apples on a local market, each apple they sell costs 15 pesos . How much it will cost you to buy 10 apples ? Then construct your function as that can give you what is required by the problem. Where   The function when evaluated can give you the total cost of 10 apples when the input is the number of goods you have to buy to be multiplies by the price of each apple sold.

19. When this function is evaluated: Where ↓ ↓   On the solution shown, the total cost of 10 apples bought will be 150 pesos . Make sure to provide appropriate unit on the result. Not every word problem will result to this function. You have to check and comprehend the problem for you to construct an appropriate solution.

20. Another Example An average household contributes an average of 1000 pesos of taxes to the government every month this year. How much will be the total tax contribution of a neighborhood with 25 families if the government declares a tax subsidy of half of the current collectible per household every month ? On this problem, the function will be: Where:  

21. When this function is evaluated: Where: ↓ ↓   So on this solution, with 25 families and a subsidy applied , the government can collect an average of 12500 pesos on that neighborhood per month.

22. Basic Forms of Functions Functions can be in various types and forms, each of which can have a distinct characteristics on the construction of equation and on the results made.       On the right are examples of the basic forms of functions; linear, quadratic, and piecewise.

23. Linear Functions These are functions which includes terms which a maximum degree of 1 . When evaluated, results to consistent intervals when a same interval of input values were applied. ↓ As evaluated, there is an interval of 2 units on the output value per interval of the input value .  

24.   Values generated by this type of function will result to a linear graph.

25. Example Amber gains 50 Primogems by doing daily quests. How much Primogems will she have if she skips spending it for 20 days, 40 days, and 60 days ? On this problem, the function will be: Whereas is the number of days.   The problem also requires to find the output for every 20 day interval until 60 days .

26. if   if   if   Looking at the results, the gap between each of the outputs have a uniform interval of 1000 which shows a characteristic of a linear function.

27. Quadratic Functions These functions includes terms with a maximum degree of 2 . When evaluated, results to intervals with output values doubled compare to the previous input value. ↓  

28.   The output values from this quadratic equation seem to be increasing in interval when the input value increases . This generates a parabolic graph.

29. Example A floor tile can cover 2 ft square of floor area. How much floor tiles are required to cover a 4 by 4, 8 by 8, and a 12 by 12 ft floor area? On this problem, the function will be: ↓ Where is the dimensions of the floor to be covered   Since the input values were the same per required output , when evaluated gives you a variable in the 2 nd degree .

30. if   Looking at the results, despite of the uniform interval on the input values , the output values exhibits exponential changes . if   if  

31. Piecewise Functions Consists of multiple expressions which are assigned on certain conditions . When a condition was met on one of the expression, that expression will be used to acquire the output. ↓ if if  

32. An common example of usage of this type of function in on trade where you set conditional prices on specific amount of goods bought just like the problem provided: A fish vendor sells Tilapia for 150 pesos per kilo but you get a 20% discount on the total purchase if you buy above 3 kilos. How much will it cost you if you buy 5 kilos?

33. The problem will lead to this function:   Whereas is identified as the weight of the fish to buy.   The first expression is used when the weight of the purchase is below 3 kilos The second expression is used when the weight of the purchase is 3 kilos and above An equation for a discount is used on the second expression which uses the total purchase and multiplied to the percentage of discount which is later deducted on the total purchase.

34. Since the problem indicates that 5 kilos is the amount to purchase (the value of x) ; This expression is what were going to use in order to find the total cost of purchase .   Where