2. Agenda Week 10 Lecture coverage: –Functions –Types of Function –Composite function –Inverse of a function

3. Functions

4. INTRODUCTION TO FUNCTIONS A function is an association of exactly one object from one set (the range) with each object from another set (the domain). This means there must be at least one arrow leaving each point in the domain. Also that there can be no more than one arrow leaving each point in the domain

5. ELEMENTS OF A FUNCTION We write to indicate that f is a function from A to B. The set A is called the domain of f. The set B is called the co-domain of f. The range of f denoted by f [A], is the set of all images; that is, The pre-image or inverse image of a set B contained in the range of f is denoted by and is the subset of the domain whose members have images in B.

6. Examples

7. Examples (Cont.)

8. Examples (Cont.) Example : Neither of the diagrams provide proper function definitions since (i) f(b) is not defined (ii) f(c) is not uniquely defined.

9. Examples (Cont.)

10. Using Congruence to Define Functions

11. Using Congruence to Define Functions (cont.)

12. Types of function …

13. INJECTIONS Let be a function. The function f is called an injective function, or an injection, if implies x = y Graphically this means that if two arrows arrive at the same point in B, they must come from the same point in A, and therefore they are the same. An injective function is also called a one-to-one function, or a function

14. INJECTIONS Example : Graphically represent an injective function one-to-one function

15. SURJECTIONS The function f is called a surjective function, or a surjection, if for each Graphically this means there must be an arrow arriving at each point of B. A surjective function is also called an onto function. If the co domain set is equal to range set then the function is surjective or on to.

16. SURJECTIONS Example : Graphically represent a surjective function onto function

17. BIJECTIONS A function can also be neither 1 -1 nor onto, or it can be both 1 -1 and onto. If a function is both 1 - 1 and onto it is called a bijection or bijective function.

18. BIJECTIONS Example : Graphically represent a bijective function or

19. BIJECTIONS Example : Graphically represent a function that is neither 1 - 1 nor onto or

20. Some Facts.. A graph of a function f is 1 - 1 iff every horizontal line intersects the graph in at most one point. A graph of a function f is onto iff every horizontal line intersects the graph in at least one point. The codomain and the range are equivalent iff the function is onto.

21. Identity Function

22. Identity Function (cont.)

23. Composite Function

24. COMPOSITE FUNCTIONS As functions are subsets of relations, the composition of a function is the same as for relations.The composition of two functions f and g is denoted by f o g(x). Example : If f(x)=2x +3 and g(x) = x+2 find fog(x).

25. Function composition (Cont.)

26. INVERTIBLE FUNCTIONS Any function f has an inverse relation, The inverse relation does not need to be a function. If the inverse relation of a function is a function, we say that the function is invertible. Steps to find the inverse of any function y = f(x) 1.Find the value of x in terms of y. 2.Interchange x and y. 3.The result of step 2 will be the inverse of the given function Example: Find the inverse of the function f(x) = 4x – 1

27. The End