Functions and Limit

A function is a rule or correspondence which associates to each number x in a set A a unique number f(x) in a set B. The set A is called the domain of f and the set of all f(x)'s is called the range of f. Definition of Functions

Let A and B be any two non-empty sets. Then a function f is a rule from A to B which associate to each element ‘a’ of A to a unique element ‘b’ of B. The element ‘b’ is called IMAGE of ‘a’ under f. The element ‘a’ is called PRE-IMAGE of ‘b’. Set A is called DOMAIN of f and B is called CODOMAIN of f.

DOMAIN RANGE X Y f x x x y y

M : Mother Function Joe Samantha Anna Ian Chelsea George Laura Julie Hilary Barbara Sue HumansMothers

M: Mother function Domain of M = {Joe, Samantha, Anna, Ian, Chelsea, George} Range of M = {Laura, Julie, Hilary, Barbara} In function notation we write M(Anna) = Julie M(George) = Barbara M(x)=Hilary indicates that x = Chelsea

For the function f below, evaluate f at the indicated values and find the domain and range of f f(1) f(2) f(3)f(4) f(5)f(6) f(7) Domain of f Range of f

Example Let A = {a,b,c} and B ={x,y,z,t}. The diagrams in the figures given below show correspondence by which elements of A are associated to elements of B AabcAabc BxyZtBxyZt (Fig.1) In Fig.1, each element of A is associated to a unique element of B.

AabcAabc BxyZtBxyZt (Fig2) In Fig.2, element ‘a’ of A is mapped to two elements x and y of B which is against the definition of a function (as per definition, to each element of A there must be a unique element of B.) So Fig.2 doesn’t define a function.

AabcAabc BxyZtBxyZt AabcAabc BxyZtBxyZt (Fig.3) Fig.4) In Fig.3, element ‘c’ is not mapped to any element of B. So this does not define a function. In Fig.4, Each element of A is mapped to a unique element of B. (This defines a function A to B)

Find Range of f from the following figure? A123A123 BabcdBabcd Domain f = set of elements of A = {1,2,3} Co domain f = Set of elements of B={a, b, c, d} Range f = {a, c} f(1) = a f(2) = a f(3) = c

Algebra of Functions If ‘f’ and ‘g’ are two functions then (1) (f+g)(x) = f(x) +g(x) (2)(fg)(x) = f(x) g(x) (3) (f-g)(x) = f(x)-g(x) (4)(f/g)(x) = f(x) / g(x), where g(x) # 0

Evaluating functions

Graph of a function The graph of the function f(x) is the set of points (x,y) in the xy-plane that satisfy the relation y = f(x). Example: The graph of the function f(x) = 2x – 1 is the graph of the equation y = 2x – 1, which is a line.

Domain and Range from the Graph of a function Domain = {x / or } Range = {y / or }

Types of Functions or Mapping f: A to B INTO Function or Mapping: If at least one element of B is not the image of an element of A, i.e. all the elements of B are not the images, then function f is called INTO function. ONTO Function or Mapping: If every element of B is image of some element of A, then f is called an ONTO function. NOTE:- EVERY MAPPING IS EITHER INTO OR ONTO.

Types of Functions or Mapping f: A to B MANY-ONE Mapping or Function: If more than one different elements of A have the same image in B, then the mapping or function f is called many one function or mapping. ONE-ONE Mapping or Function: If different elements of set A have different images, i.e. no two different elements have the same images, then function or mapping is called a one-one mapping.

Types of Functions or Mapping f: A to B Constant Function or mapping: A mapping of function f: A to B is said to be a constant mapping if all the elements of A have same image. Even and Odd Functions: A function f: A to B is said to be an even if f(-x) = f(x) and odd if f(-x) = -f(x). Equal Functions: Two functions or mappings f & g having the same domain A are said to be equal if under both the functions all the elements have same images, i.e. f(x)=g(x).

Various Functions Constant Function: A function of the form f(x) = c, where c is a constant is called a constant function. Linear function: A function of the form f(x) = ax+b, where a & b are constants and a#0 is called a linear function. Quadratic function: A function of the form, where a, b & c are constants and a#0 is called a quadratic function.

Various Functions Polynomial function: A function of the form, where a 0, a 1,….a n are constants and n is non- negative integer, a 0 #0, is called a polynomial function of x of degree n. Rational function: A function of the form f(x) = p(x) / q(x), q(x) #0 & where p(x) & q(x) are polynomialis called a Rational function. Exponential function: A function of the form, where a>0 & a#1, x is any real number is called exponential function.

x

Let the demand be 40 units of a product when the price is Rs. 10 per unit and 20 units when price is Rs. 15 per unit. Find the demand function assuming it to be linear. Also determine the price when demand is 16 units. Solution: As Demand is a linear function P = a + b X where, P is price per unit & X is quantity demand

Now P = 10 when X = 40 implies that 10 = a + 40 b …….(1) Also, at X= 20, P = 15 we have 15 = a + 20 b………. (2) Solving (1) and (2), we get a= 20, b = -1/4 Therefore Demand function is p = 20 – (¼ )X …………………… (3) By putting X = 16 in (3), We get P = 16 ( Price is Rs. 16 per unit when demand is 16 units)

1. One-one function There is one-one correspondence between the elements of the set A and the set B. 2. Many-one function There is many-one correspondence between the elements of the set A and the set B. 3. Onto function Every element of the set B has at least one pre-image. In above fig.(i), the function is one-one and onto, while in fig.(ii) the function is many-one and onto. 4. Into function There is at least one element of B which has no pre-image.

One to one MANY TO ONE ONTO INTO

LIMITS

Introduction to Limits The student will learn about: Functions and graphs, limits from a graphic approach, limits from an algebraic approach, and limits of difference quotients.

Functions and Graphs A Brief Review The graph of a function is the graph of the set of all ordered pairs that satisfy the function. As an example the following graph and table represent the same function f (x) = 2x – 1. xf(x) ? 4? We will use this point on the next slide.

Limits (THIS IS IMPORTANT) Analyzing a limit. We can also examine what occurs at a particular point by the limit ideas presented in the previous chapter. Using the above function, f (x) = 2x – 1, let’s examine what happens when x = 2 through the following chart: x f (x) ? Note; As x approaches 2, f (x) approaches 3. This is a dynamic situation

Limits IMPORTANT! This table shows what f (x) is doing as x approaches 2. Or we have the limit of the function as x approaches 2. We have written this procedure with the following notation. x f (x) ? Def: We write if the functional value of f (x) is close to the single real number L whenever x is close to, but not equal to, c. (on either side of c). or as x → c, then f (x) → L 2 3

One-Sided Limit This idea introduces the idea of one-sided limits. We write and call K the limit from the left (or left-hand limit) if f (x) is close to K whenever x is close to c, but to the left of c on the real number line.

One-Sided Limit We write and call L the limit from the right (or right-hand limit) if f (x) is close to L whenever x is close to c, but to the right of c on the real number line.

The Limit Thus we have a left-sided limit: And a right-sided limit: And in order for a limit to exist, the limit from the left and the limit from the right must exist and be equal.

Example 1 Example from my graphing calculator. The limit does not exist at 2! The limit exists at

Limit Properties Let f and g be two functions, and assume that the following two limits are real and exist. the limit of the sum of the functions is equal to the sum of the limits. Then: the limit of the difference of the functions is equal to the difference of the limits.

Limit Properties If: the limit of the nth root of a function is the nth root of the limit of that function. the limit of the quotient of the functions is the quotient of the limits of the functions. The limit of a constant times a function is equal to the constant times the limit of the function. Then: the limit of the product of the functions is the product of the limits of the functions.

Example 2 x 2 – 3x =x 2 - 3x = 4-2. From this example we can see that for a polynomial function f (x) = f (c) – 6 =

Indeterminate Form The term indeterminate is used because the limit may or may not exist. If and, then is said to be indeterminate.

The Limit of a Difference Quotient. Let f (x) = 3x –1. Find f (a + h) =3 (a + h) - 1 =3a + 3h - 1 f ( a ) = 3 ( a ) – 1 = 3a - 1 f (a + h) – f (a) = 3h 3

Limit Theorems

Examples Using Limit Rule Ex.

More Examples

Indeterminate forms occur when substitution in the limit results in 0/0. In such cases either factor or rationalize the expressions. Ex. Notice form Factor and cancel common factors Indeterminate Forms

More Examples

Summary. We now have some very powerful tools for dealing with limits and can go on to our study of calculus. Please pay special attention to the practice problems. We started by using a table to see how a limit would react. This was an intuitive way to approach limits. That is they behaved just like we would expect. We saw that if the left and right limits were the same we had a limit. We saw that we could add, subtract, multiply, and divide limits.

Limits at Infinity For all n > 0, provided that is defined. Ex. Divide by

More Examples

Practice Problems

Evaluate the following limits

Def.: A function f(x) is said to tend to a number L as x approaches to c if to a given there exist a Such that and we write Limit of a function exists if Right Hand Limit = Left Hand Limit

Algebra of Limits