E- Content in Mathematics

Algebra and Trigonometry 3

E- Content in Mathematics

FUNCTIONS

Objectives

From this unit a learner is expected to achieve the following

1. Learns the concept of function from a set to another set.
2. Familiarizes with various terms domain, target set, range etc.
3. Familiarizes with pictorial representation of functions between sets.
4. Familiarizes withcomposition offunctions.
5. Understands that composition of functions is associative.
6. Familiarizes with one-to-one, onto and invertible functions.

Sections

1. Functions
2. Functions as Relations
3. Composition of Functions
4. One-to-one, Onto and Invertible Functions
5. Geometrical Characterization of One-to-one and Onto Functions

In this session we introduce the concept of a function from one set into another. Without exaggeration this is probably the single most important and universal notion that runs through all of mathematics.

1. Functions

Suppose that to each element of a set A we assign a unique element of a set Baccording to certain rule; thentherule by which the assignments are madeis called a function from A into B.

DefinitionLet A and B be two non-empty sets. A function from A to B is a rule which assigns to each element of A a unique element of B. A function is also called map or mapping or transformation. The set A is called the domain of the function, and the set B is called the target set(or codomain).

Functions are ordinarily denoted by symbols.

• Let denote a function from A into B. Then we write

which is read: “f is a function from A into B”, or “ takes A into B”, or “ maps A into B”.

• Suppose and . Then will denote the unique element of B which f assigns to a. This element in B is called the image of under or the value of f at a. We also say that f sends or maps into. The set of all such image values is called the range or image of f , and it is denoted by Ran (f), or Im (f) or . That is

for some }

• Obviously, is a subset of the target set B.
• Usually, a function isexpressed bymeans of a mathematical formula: Consider the function which sends each real number into its square. We may describe this function by writing

In the first notation, x is called an argument (variable) and the letter f denotes the function. In the second notation, the barred arrow is read goes into. In the last notation, x is called the independent variable and y is called the dependent variable since the value of y will depend on the value of x.

• Furthermore, suppose a function is given by a formula in terms of a variable x. Then we assume, unless otherwise stated, that the domain of the function is the set of real numbers or the largest subset of for which the formula has meaning and that the codomain is .
• Suppose . If is a subset of , then denotes the set of images of elements in and if is a subset of , then denotes the set of elements of A each whose image belongs to That is,

We call the image of , and the inverse image or preimage of .

Example 1Consider the function , i.e., assigns to each real number its cube. Then the image of 2 is 8, and so we may write . Similarly, , and .

Example 2 Let g assign to each country in the world its capital city. Here the domain of g is the set of all the countries in the world, and the codomainis theset of cities in the world. The image of France under g is Paris; that is g(France)= Paris. Similarly, g(India) =New Delhi and g (England) = London.

Example 3Find the domain D of each of the following real-valued functions:

Solution

• is not defined for ; hence
• g is defined for every real number; hence D = .
• h is not defined when is negative; hence .Example 4Suppose. Find the numberof functions from A into B and from B into A.

Solution

• There are three choices, 1,2 or 3 for the image of a, and similarly, three choices for the image of b. Hence there are functions from A into B.
• There are two choices, a or b, for each of the three elements of B. Hence there are functions from B into A.

Example 5Suppose and are finite sets with elements and elements, respectively. Show that there are functions from into .

Solution

There are |B| choices for each of the |A| elements of A; hence there are possible functions

from A into B.

The set of all functions from into is usually denoted by. With this notation,by Example 5, it is clear that if and are finite sets then

Identity Function

Consider any set . Then there is a function from into which sends each elementof A into itself. It is called the identity function on and it is usually denoted by or or simply 1. In otherwords, the identity function is defined by

for every element .

Functions as Relations

DefinitionConsider the function . Then

Graph of

Remark Note that Graph of is a subset of the Cartesian product . Hence, every function gives rise to a relation from A to B called the

graph of f. We do not distinguish between a function and its graph. The relation graph of the function has the property that each a in Abelongs to a unique ordered pair in the relation. On the other hand, any relation from A to B that has this property gives rise to a function , where for each in f . Consequently, one may equivalently define a function as follows:

Definition A function is a relation from A to B such that each is the first component of a unique ordered pair in f.

Example 6Consider the following relations on the set :

• f is a function from A into A since each member of A appears as the first coordinate in exactly one ordered pair in f; here .
• g is not a function from A into A since is not the first coordinate of any ordered pair in g.
• h is not a function from A into A since appears as the first coordinate of two distinct ordered pairs (1, 3) and (1, 2) in h.

Definition Two functions and are said to be equal, if for every ; that is, if and have the same graph. If is equal to we write .

Pictorial Representation of Functions

Although we do not distinguish between a function and its graph, we will still use the terminology “graph of ” when referring to as a set of ordered pairs. Moreover, since the graph of is a relation, we can draw its pictures(such as arrow diagram andplotting the points in a plane) as was done for relations in general, and this pictorial representation itself is sometimes called the graph of .

Example7The arrow diagram Fig.1 defines a function f from A = {a, b, c, d} into B = {r, s, t, u} in the obvious way; that is

The image of f is the set {r, s, u}. Note that t does not belong to the image of f because t is not the image of any element of A under f .

Fig. 1

Example 8Let and let be defined by the diagram in Fig. 2

• Find the graph of as a set of ordered pairs.
• Findthe image of .
• Find where .
• Find where

Fig. 2

Solution

• The graph of consists of all ordered pairs where. Hence

• The image of isand it consists of all image points. Since only 2,3, and 5 appear as image points
• .
• The element 4 has image 2, and the elements 1 and 5 have image 3; hence

Plotting the graph of a polynomial

A real poynomial functionis a function of the form

where the are real numbers. Since is an infinite set, it would be impossible to plot each point of the graph. However, the graph of such a function can be approximated by first plotting some of its points and then drawing a smooth curve through these points. The points are usually obtained from a table

where various values are assigned to x and the corresponding values of f (x) are computed.

Example 9 Fig. 3 illustrates the above technique using the function .

Fig. 3

Vertical Line Test

The defining condition of a function from to , that is the first component of a unique ordered pair in , is equivalent to the geometrical condition that each vertical line intersects the plotted graph at exactly one point. Hence, a set of points in the plane is a function if and only if every

vertical line contains exactly one point of the set.

Example 10Determine which of the graphs in Fig. 4 are functions from into .

Fig. 4

Solution

• The graph in Fig. 4(a) is a function.
• The graph in Fig. 4(b) is not a function since any vertical line to the right of meets the curve at more than one point.
• The graph in Fig. 4(c) is not a function, since any vertical line to the right of doesn’t meet the curve at any point. However the graph does define a function from D into where

Composition of Functions

Consider the functions and where the codomainB of f is the domain of g. This relationship can be pictured by the following diagram:

Let ; then its image under f is in B which is the domain of g. Accordingly, we can find the image of under the function g, that is, we can find . Thus we have arule which assigns to each element a in A an element in C or, in other words, f and g give rise to a well defined function from A to C. The new function is called the composition of f and g, and is denoted by That is, if and , then we define a new function by

for every a in A.

The functions f,g and can be diagrammatically represented as follows:

Attention! We emphasize that the composition of f and g is written , and not ; that is, the composition of functions is read from right to left, and not from left to right.

Example11 Let and be the functions defined by Fig.5. We compute : by its definition:

.

Similarly, it can be seen that

,

and

The above can also be obtained by “following the arrows” from A to C in the diagram of the functions f and g.

Fig. 5

Example12 Consider any function . Then one can easily show that

where are the identity functions on A and B, respectively. In other words, the composition of any function with the appropriate identity function is the function itself.

Associative Property of Composition of Functions

Fig. 6

Consider the functions , and . Then, as pictured in Fig. 6(a), we can form the composition , and then the composition . Similarly, as pictured in Fig.6(b), we can form the composition , and then the composition. Both are functions with domain A and target set D. The next theorem on

functions states that these two functions are equal.

Theorem 1 Let ,, and . Then

The proof of the theorem is left to the assignments.

Remark In view of Theorem 1 we can write without any parentheses.

ONE TO ONE, ONTO AND INVERTIBLE FUNCTIONS

Definition A function is said to be one-to-one (written ) if implies .

That is, is 1-1 if different elements in the domain A have distinct images.

Definition A function is said to be an onto function if

.

That is, is onto if every element of B is the image of some elements in A or, in other words, if the image of f is the entire target set B. In such a case we say that f is a function of A onto B or that f maps A onto B.

Remarks

• The term injective is used for a one-to-one function, surjective for an onto function, and bijective for afunction which is one-to- one and onto.
• If is both one-to-one and onto, thenf is called a one-to-one correspondence between A and B. This terminology comes from the fact that each element of A will correspond to a
• unique element of B and vice versa.

Example 13Letbe a non-empty set. The identity function is a one-to-one and onto function and hence is bijective.

Example 14Consider functions and . Prove the following.

(a)If is one-to-one, then is one-to-one.

(b)If is onto, then is onto.

Solution

(a)Suppose is not one-to-one. Then there exist distinct elements for which . Thus

;

hence is not one-to-one. Therefore, if is one-to-one, then must be one-to-one.

(b)If then ; hence . Suppose g is not onto. Then is properly contained in C and so is properly contained in C; thus is not onto, then g must be onto.

Definition A function is called a constant function if assigns each

to a fixed element in .

Definition A function is said to be invertible if its inverse relation is a function from B to A.

Equivalently, is invertible if there exists a function called the inverse of f, such that

and

Attention! Suppose and if is a subset of , then we can discuss the inverse image even if is not a function.

Theorem 2 A function is invertible if and only if f is both one-to-one and onto.

Proof

Suppose is invertible. Then there exists a function for which and Since is one-to-one, is one to one, and hence by Example 14 f is one-to-one; and since is onto, is onto, and hence by Example 14 is onto. That is, is both one-to-one and onto.

Now suppose is both one-to-

one and onto. Then each is the image of a unique element in A, say . Thus if f (a) = b, then a = ; hence f () = b. Now let denote the mapping from to defined by g (b) = . We have

Hence

Hence

Hence by the definition, f is invertible and its inverse is the mapping g.

Geometrical Characterization of One-to-one and Onto Functions

We now discuss the geometrical meaning of the concepts of being one-to-one and onto of a real valued function . We first note that f may be identified with its graph and the graph may be plotted in the Cartesian plane .

• The function is one-to-one means that there are no two distinct ordered pairs in the graph of f: hence each horizontal line in can intersect the graph of f in at most one point.
• The function is onto means that for every there is at least one point such that (a, b) belongs to the graph of f: hence each horizontal line in must intersect the graph of f at least once.
• Hence the function is one-to-one and onto, i.e., f is invertible, if and only if each horizontal line in will intersect the graph of f in exactly one point.

We illustrate the above properties in the next example.

Example15 Consider the following four functions from into whose graphs appear in Fig. 7:

Fig. 7

Observe that there are horizontal lines which intersect the graph of twice and there are horizontal lines which do not intersect the graph of at all; hence is neither one-to-one nor onto. Similarly, is one-to-one but not onto, and is onto but not one-to-one. But is both one-to-one and onto. The inverse of is the cube root function given by

Remark Sometimes we restrict the domain and codomain of a function in order to obtain an inverse function . For example, suppose we restrict the domain and codomain of the function to be the set D of nonnegative real numbers. Then is one-to-one and onto and its inverse is the square root function given by

Similarly, suppose we restrict the codomain of the exponential function to be the set of positive real numbers. Then is one-to-one and onto and its inverse is the logarithmic function (to the base 2) given by

For a non-empty set we denote the set of all one-to-one and onto mappings of onto itself by We state some properties of in the following theorem. The proof is left to the assignments.

Theorem 3

If are elements of then

1. is in
2. The identity function is an element in and .
3. There exists an element in such that

Theorem 4 If S has more than two elements, we can find two elements in such that

Proof. Let be three distinct elements of S. Define the mapping

by

and for any different from

Define the mapping

by

and for any different from

Clearly both are in . Now

and

and hence

Summary

In this session we have discussed the concept of function between two sets. We have seen that function is a special type of relation. We have discussed various ways of representing functions. We have observed the associativity of composition of functions. The concepts of one-to-one, onto and invertible functions are also discussed.

Assignments

1.Let A be the set of students in a school. Determine which of the following assignments defines a function on A:

(a) To each student assign his age.

(b) To each student assign his teacher.

(c) To each student assign his sex.

(d) To each student assign his spouse.

2. Define each of the following functions from into by a formula:

(a)To each number let assign its square plus 3.

(b) To each number let g assign its cube plus twice the number.

(c)To each number greater than or equal to 3 let h assign the square of the number; and to each number less than 3 let h assign the number –2.

3. Let be defined by

Find .

4. Let the function be defined as follows:

Find:

5. Let . Find the number of functions from A into B and from B into A.

6. Find the number of constant functions from A into B.

7. Let . Determine whether or not each relation below is a function from X into X.

(a)

(b)

(c)

8.Let the functions and be defined by Fig. 8. Find the composition function .

Fig. 8.

9. Let the function and be defined by and . Find the formula defining the composition functions: (a) (b)

10. Determine whether each of the following functions are one-to-one.

(a)To each person on the earth assign the number which corresponds to his age.

(b)To each country in the world assign the latitude and longitude of its capital.

(c)To each state in India assign the name of its capital.

(d)To each book written by only one author assign the author.

(e)To each country in the world which has a prime minister assign its prime minister.

11. Let the functions be defined by Fig. 9.

(a)Determinewhether each of the following functions is one-to-one.

(b)Determine whether each function is onto.

(c)Determine whether each function is invertible.

(d)Find the composition

Fig. 9.

12.Consider functions . Prove the following.

(a)If and are one –to-one, then the composition function is one-to-one.

(b)If and are onto functions, then is an onto function.