1. Relations And Functions

2. A relation from non empty set A to a non empty set B is a subset of cartesian product of A x B. This is a relation The domain is the set of all x values in the relation The range is the set of all y values in the relation {(2,3), (-1,5), (4,-2), (9,9), (0,-6)} domain = {-1,0,2,4,9} These are the x values written in a set from smallest to largest range = {-6,-2,3,5,9} These are the y values written in a set from smallest to largest

3. Domain (set of all x’s) Range (set of all y’s) 1 2 3 4 5 2 10 8 6 4 A relation assigns the x’s with y’s This relation can be written {(1,6), (2,2), (3,4), (4,8), (5,10)} independent values

4. A function f from set A to set B is a rule of correspondence that assigns to each element x in the set A exactly one element y in the set B. Whew! What did that say? Set A is the domain 1 2 3 4 5 Set B is the range 2 10 8 6 4 A function f from set A to set B is a rule of correspondence that assigns to each element x in the set A exactly one element y in the set B. Must use all the x’s A function f from set A to set B is a rule of correspondence that assigns to each element x in the set A exactly one element y in the set B. The x value can only be assigned to one y This is a function ---it meets our conditions All x’s are assigned No x has more than one y assigned

5. Set A is the domain 1 2 3 4 5 Set B is the range 2 10 8 6 4 Must use all the x’s Let’s look at another relation and decide if it is a function. The x value can only be assigned to one y This is a function ---it meets our conditions All x’s are assigned No x has more than one y assigned The second condition says each x can have only one y, but it CAN be the same y as another x gets assigned to.

6. A good example that you can “relate” to is students in our maths class this semester are set A. The grade they earn out of the class is set B. Each student must be assigned a grade and can only be assigned ONE grade, but more than one student can get the same grade (we hope so---we want lots of A’s). The example show on the previous screen had each student getting the same grade. That’s okay. 1 2 3 4 5 2 10 8 6 4 Is the relation shown above a function? NO Why not??? 2 was assigned both 4 and 10 A good example that you can “relate” to is students in our maths class this semester are set A. The grade they earn out of the class is set B. Each student must be assigned a grade and can only be assigned ONE grade, but more than one student can get the same grade (we hope so---we want lots of A’s). The example shown on the previous screen had each student getting the same grade. That’s okay.

7. Set A is the domain 1 2 3 4 5 Set B is the range 2 10 8 6 4 Must use all the x’s The x value can only be assigned to one y This is not a function---it doesn’t assign each x with a y Check this relation out to determine if it is a function. It is not---3 didn’t get assigned to anything Comparing to our example, a student in maths must receive a grade

8. Set A is the domain 1 2 3 4 5 Set B is the range 2 10 8 6 4 Must use all the x’s The x value can only be assigned to one y This is a function Check this relation out to determine if it is a function. This is fine—each student gets only one grade. More than one can get an A and I don’t have to give any D’s (so all y’s don’t need to be used).

9. Vertical Line Test for a Function A set of points in a coordinate plane is the graph of y as a function of x if and only if no vertical line intersects the graph at more than one point.

10. On the interval containing x 1 < x 2, 1. f(x) is increasing if f(x 1 ) < f(x 2 ). Graph of f(x) goes up to the right. 2. f(x) is decreasing if f(x 1 ) > f(x 2 ). Graph of f(x) goes down to the right. On any interval, 3. f(x) is constant if f(x 1 ) = f(x 2 ). Graph of f(x) is horizontal. Increasing, Decreasing, and Constant Function

11. We commonly call functions by letters. Because function starts with f, it is a commonly used letter to refer to functions. The left hand side of this equation is the function notation. It tells us two things. We called the function f and the variable in the function is x. This means the right hand side is a function called f This means the right hand side has the variable x in it The left side DOES NOT MEAN f times x like brackets usually do, it simply tells us what is on the right hand side.

12. So we have a function called f that has the variable x in it. Using function notation we could then ask the following: Find f (2). This means to find the function f and instead of having an x in it, put a 2 in it. So let’s take the function above and make brackets everywhere the x was and in its place, put in a 2. Don’t forget order of operations---powers, then multiplication, finally addition & subtraction Remember---this tells you what is on the right hand side---it is not something you work. It says that the right hand side is the function f and it has x in it.

13. Find f (-2). This means to find the function f and instead of having an x in it, put a -2 in it. So let’s take the function above and make brackets everywhere the x was and in its place, put in a -2. Don’t forget order of operations---powers, then multiplication, finally addition & subtraction

14. Find f (k). This means to find the function f and instead of having an x in it, put a k in it. So let’s take the function above and make brackets everywhere the x was and in its place, put in a k. Don’t forget order of operations---powers, then multiplication, finally addition & subtraction

15. Find f (2k). This means to find the function f and instead of having an x in it, put a 2k in it. So let’s take the function above and make brackets everywhere the x was and in its place, put in a 2k. Don’t forget order of operations---powers, then multiplication, finally addition & subtraction

16. Let's try a new function Find g(1)+ g(-4).

17. The last thing we need to learn about functions for this section is something about their domain. Recall domain meant "Set A" which is the set of values you plug in for x. For the functions we will be dealing with, there are two "illegals": 1.You can't divide by zero (denominator (bottom) of a fraction can't be zero) 2.You can't take the square root (or even root) of a negative number 3. Practical problems may limit domain. When you are asked to find the domain of a function, you can use any value for x as long as the value won't create an "illegal" situation.

18. Find the domain for the following functions: Since no matter what value you choose for x, you won't be dividing by zero or square rooting a negative number, you can use anything you want so we say the answer is: All real numbers x. If you choose x = 2, the denominator will be 2 – 2 = 0 which is illegal because you can't divide by zero. The answer then is: All real numbers x such that x ≠ 2. means does not equal illegal if this is zero Note: There is nothing wrong with the top = 0 just means the fraction = 0

19. Let's find the domain of another one: We have to be careful what x's we use so that the second "illegal" of square rooting a negative doesn't happen. This means the "stuff" under the square root must be greater than or equal to zero (maths way of saying "not negative"). Can't be negative so must be ≥ 0 solve this So the answer is: All real numbers x such that x ⋲ [4, ∞)

20. Summary of How to Find the Domain of a Function Look for any fractions or square roots that could cause one of the two "illegals" to happen. If there aren't any, then the domain is All real numbers x. If there are fractions, figure out what values would make the bottom equal zero and those are the values you can't use. The answer would be: All real numbers x such that x ≠ those values. If there is a square root, the "stuff" under the square root cannot be negative so set the stuff < 0 and solve. Then answer would be: All real numbers x such that x ≠ whatever you got when you solved. NOTE: Of course your variable doesn't have to be x, can be whatever is in the problem.

21. Real function A function which has either R or one of its subsets as its range is called a real valued function. In the function if domain is also either R or a subset of R, it is called a real function. Real valued function

22. Some functions and their graphs Identity function- Let R be the set of real no. define the real valued function f: R R BY y= f(x) for each x ⋲R such a function is called identity function. Here the domain and range of f are R. Its graph passes through the origin.

23. As visible above, the graph of the identity function consists of a 45 o line through the origin. Any point of the identity function may be written as (x, x) since f(x) = x.

24. Modulus function The function f:R R defined by y=f(x) = |x|for each x ⋲R is called a modulus function. For each non negative value of x, f(x) is equal to x. But for a non negative value of x, the value of f(x) is negative of the value of x. such a function is called Piece-Wise Defined Function The Absolute value of a number x is written |x| and is defined as |x| = x if x ≥ 0 or |x| = −x if x < 0.

26. Constant function Define the function f: R R by y =f(x)=c, x ⋲R where c is a constant and x ⋲R here domain of f is R and its range is {c}. The graph is a line parallel to x-axis. For ex- f(x)=3 for each x ⋲R is a constant function.

28. Polynomial Function Let n be a nonnegative integer and let a n, a n-1,…, a 2, a 1, a 0, be real numbers with a n  0. The function defined by f (x)  a n x n  a n-1 x n-1  …  a 2 x 2  a 1 x  a 0 is called a polynomial function of x of degree n. The number a n, the coefficient of the variable to the highest power, is called the leading coefficient.

29. The graph of f ( x ) = x 3 − x 2 − 4 x + 4.

30. Greatest integer function The function f:R R defined by y= f(x) = [x], x⋲R assumes the value of the greatest integer, less than or equal to x Such a function is called signum functions

32. Solved Example on Greatest Integer Function Find: (a) [ -256 ] (b) [ 3.506 ] (c) [ -0.7 ] Solution: By the definition of greatest integer function, (a) [ -256 ] = -256 (b) [ 3.506 ] = 3 (c) [ -0.7 ] = -1

33. Rational functions Definition A rational function f has the form where g (x) and h (x) are polynomial functions. The domain of f is the set of all real numbers except the values of x that make the denominator h (x) zero. In what follows, we assume that g (x) and h (x) have no common factors.

35. Signum function The greatest integer function (or floor function) will round any number down to the nearest integer. The notation for the greatest integer function is shown f(x) = 1, if x>0 0, if x=0 -1 if x<0

36. Signum function

37. Try these 1. Let A= { 1,2,3,………14}. Define a relation R from A to A by R= { (x,y): 3x-y=0, where x,y ⋲A} write down its domain, codomain and range.

38. If f(x)= find 1. f(-2) 2. f(1) 3. f(3) X 2 when x<0 X when 0<x<1 1 /x when x>1

39. Which of the following relations are functions? Give reasons If it is a function determine its range. 1. f(x)= {(x,y): y=x+1)} 2. f(x)= {(x,y): x+y>4 )} 3. f(x)= {(x,y): x+y=5 )}

40. Name- vaishali agarwal class – xi a roll no. 40