Operation on Functions.pptx

general math module 3

1. Operation on Functions

2. Operations on functions are similar to operations on numbers. Adding, subtracting and multiplying two or more functions together will result in another function. Dividing two functions together will also result in another function if the denominator or divisor is not the zero function. Lastly, composing two or more functions will also produce another function.

3. ACTIVITY: SECRET MESSAGE Direction. Answer each question by matching column A with column B. Write the letter of the correct answer at the blank before each number. Decode the secret message below using the letters of the answers. Column A Column B _____1. Find the LCD of and . A. (x + 4)(x − 3) 3 1 4x+7 _____2. Find the LCD of x−2 and x+3 C. x 2+x−6 _____3. Find the sum of and . D. 2 5 2 + x − 6 _____4. Find the sum of + E. (𝑥 − 2)(𝑥 + 3) or x x x 𝑥+4

4. 5. Find the product of and . G. x+2 3 1 _____6. Find the sum of and H. (x + 1)(x − 6) x−2 x+3 For numbers 7-14, find the factors. _____7. x2 + x − 12 I. _____8. x2 − 5x − 6 L. (𝑥 − 4(𝑥 − 3) _____9. x2 + 6x + 5 M. −5 _____10. x2 + 7x + 12 N. 21 _____11. x2 − 7x + 12 O. (𝑥 − 5)(𝑥 − 3) _____12. x2 − 5x − 14 R. (x + 4)(x + 3) _____13. x2 − 8x + 15 S. (𝑥 − 7)(𝑥 − 5) _____14. x2 − 12x + 35 T. x2+x−12 x2+6x+5 _____15. Find the product of x2−5x−6 and x2+7x+12. U. (𝑥 − 7(𝑥 + 2) x2−5x−14 x2−12x+35 𝑥 _____17. In the function f(x) = 4 − x2, 𝑓𝑖𝑛𝑑 𝑓(−3) Y. (x + 5)(x + 1) x2+x−12 x2−8x+15 7 _____16. Divide by W.

5. Definition. Let f and g be functions. Their sum, denoted by 𝑓 + 𝑔, is the function denoted by (𝑓 + 𝑔)(𝑥) = 𝑓(𝑥) + 𝑔(𝑥). Their difference, denoted by 𝑓 − 𝑔, is the function denoted by (𝑓 − 𝑔)(𝑥) = 𝑓(𝑥) − 𝑔(𝑥). Their product, denoted by 𝑓 • 𝑔, is the function denoted by (𝑓 • 𝑔)(𝑥) = 𝑓(𝑥) • 𝑔(𝑥). Their quotient, denoted by 𝑓/𝑔, is the function denoted by (𝑓/𝑔)(𝑥) = 𝑓(𝑥)/𝑔(𝑥), excluding the values of x where 𝑔(𝑥) = 0. The composite function denoted by (𝑓 ° 𝑔)(𝑥) = 𝑓(𝑔(𝑥)). The process of obtaining a composite function is called function composition.

6. Example 1. Given the functions: 𝑓(𝑥) = 𝑥 + 5 𝑔(𝑥) = 2𝑥 − 1 ℎ(𝑥) = 2𝑥 2 + 9𝑥 − 5 Determine the following functions: (𝑓 + 𝑔)(𝑥) 𝑒. (𝑓 + 𝑔)(3) (𝑓 − 𝑔)(𝑥) 𝑓. (𝑓 − 𝑔)(3) (𝑓 • 𝑔)(𝑥) 𝑔. (𝑓 • 𝑔)(3) ( )(𝑥) h. (  

7. Solution:

8. ILLUSTRATIONS In the illustrations, the numbers above are the inputs which are all 3 while below the function machine are the outputs. The first two functions are the functions to be added, subtracted, multiplied and divided while the rightmost function is the resulting function.

9. ADDITION

10. SUBTRACTION

11. MULTIPLICATION

12. DIVISION

13. Composition of functions: In composition of functions, we will have a lot of substitutions. You learned in previous lesson that to evaluate a function, you will just substitute a certain number in all of the variables in the given function. Similarly, if a function is substituted to all variables in another function, you are performing a composition of functions to create another function. Some authors call this operation as “function of functions”.

14. b. (𝑓 ∘ ℎ)(4) = 𝑓(ℎ(4))

15. ACTIVITY: PERFORM THE INDICATED OPERATIONS a= 3 n= 3 t= 3