This document discusses various concepts relating to functions, including: 1) It defines the types of functions - surjective, injective, and bijective - and provides examples to illustrate each type. 2) It covers algebraic operations that can be performed on functions, such as addition, subtraction, multiplication, and division, demonstrating each with an example. 3) It explains the concept of function composition, using examples to demonstrate what it means for one function to be composed with another (f o g). 4) It discusses the relationship between inverse and composition functions, proving that the identity function satisfies f o f^-1 = f^-1 o f = I(x).

1. The function of composition and inversmathematic lesson From pyse group LET’S STUDY,GUYS!!!

2. The kinds of the nature of the function The function of surjective The function of injektive The function of bijektive Note: don’t memorize it!! Please,just understand it! Exercise and exercise!

3. The function of surjective Look this figure: Observe it!! A B C D E Look the next to compare!!

4. It’s not surjective Compare it with previous!! Observe the codomain and the range!!! So, what is the surjective??? A B C D E F G

5. The function of injective A B C A B C D

6. It’s not injective Compare and define!!! A B C D A B C D

7. The function of bijective A B C D A B C D

8. What is the definition??? Bijective injective+surjective. Do you understand??? I hope you understand. 

9. The algebraic operation on function Operation used is added,subtraction,multiplication,and division. We can write like this: (f+g)(x)=f(x)+g(x) (f-g)(x)=f(x)-g(x) (fxg)(x)=f(x)xg(x) (f:g)(x)=f(x):g(x) Look the example for the next!!!

10. example f(x)=3x+4 and g(x)=2x-2 Find: a)(f+g)(x)…………..b)(f-g)(x) (f+g)(x)=f(x)+g(x)= (3x+4)+(2x-2)=5x+2 (Find it yourself!) You can try other operationals!! 4points for you if you want to try

11. Composition function We will find the meaning of f o g or g o f. We can read it by f circle g or g circle f. For example that I have f(x)=x+1 and g(x)= x^2 (x square). Then , we have known that f(x)=x+1, f(1)=1+1=2, so we can change the x with g(x)! It will be f(g(x))=g(x)+1=x^2+1. and we can say h(x)=f(g(x))………………this result can be called “f circle g”……………………………and now, what is “g circle f”???? This is g circle f: Find in the next slide!!!

12. “g circle f” Ok, the way to reach is same!! f(x)=x+1 g(x)=x^2…………………………..and x must be changed by f(x) g(f(x))=(x+1)^2………………………it called g circle f!!!! Try to understand!! It’s easy!!………

13. The terms of two functions that can be composed. The intesection is g and h. Domain g Codomain f Codomain/range g ,but domain f E F I J A B C D K L M N G H Get the explain in the next slide!!!

14. explain The function can be composed if the two functions have the intersection. for example: (look the figure) g=((a,e),(b,f),(c,g),(d,h)) f=((g,k),(h,l),(I,m),(j.n)) So,we can get f o g=((c,k),(d.l)) What is the term???????? 6points for you if you understand!

15. Inverse function For example: A=(1,3,5) B=(2,4,6) And, f:AB,so C: C=((1,2),(3,4),(5,6)) determine the inverse function f, and investigate whether that inverse function is an inverse function! Invers C is f-1:BA, sof-1=((2,1),(4,3),(6,5)) because all members are mapped, then this is an inverse function.

16. Second example A =(1,2,3) B=(4,5,6,7) F:AB So, c=((1,4),(2,5),(3,6)) And , F-1:BA, so F-1=((4,1),(5,2),(6,3)) Because, not all members are mapped,then this is not an inverse function. So, we can conclude that a function has an invers function, only if f is bijective function.

17. The relation between inverse and composition function Identity function: (f o f-1)(x)=(f-1 o f)(x)=x=I(x) Can you prove it??

18. Invers of composition function The charateristic: (f o g)-1(x)=(g-1 o f-1)(x) (g o f)-1(x)=(f-1 o g-1)(x) for example: f(x)=5x+8 g(x)=x-5 Determine (f o g)-1(x)!! Get answer in the next!!!

19. answer (f o g)(x)=f(g(x))=5x-17 f(g(x))=y=5x-17 X=(y+17)/5 And X=(f o g)-1(x). Ok, it’s over!! 100points if you understand all in this chapter!!