Subsidiary Mathematics

My goals

Introduction 

A polynomial is an expression that can have constants, variables and exponents, that can be combined using addition, subtraction, multiplication and division, but: 

Polynomials are used to describe curves of various types; people use them in the real world to graph curves. 

For example, roller coaster designers may use polynomials to describe the curves in their rides. Polynomials can be used to figure how much of a garden’s surface area can be covered with a certain amount of soil. The same method applies to many flat-surface projects, including driveway, sidewalk and patio construction. Polynomials can also be used to model different situations, like in the stock market to see how prices will vary over time. Business people also use polynomials to model markets, as in to see how raising the price of a good will affect its sales. 

Functions are important in calculating medicine, building structures (houses, businesses,…), vehicle design, designing games, to build computers (formulas that are used to plug to computer programs), knowing how much change you should receive when making a purchase, driving (amount of gas needed for travel). 

Required outcomes 

After completing this unit, the learners should be able to: 

1. Generalities on numerical functions

Activity 1

In the following arrow diagram, for each of the elements of set A, state which element of B is mapped to it. 

A function is a rule that assigns to each element in a set A one and only one element in set B) We can even define a function as any relationship which takes one element of one set and assigns to it one and only one element of second set. The second set is called a co-domain. The set A is called the domain, denoted by Domf. 

If x is an element in the domain of a function f, then the element that f associates with x is denoted by the symbol 

Example 1

Four children, Ann, Bob, Card and David, are given a spelling test which is market out of 5; their marks for the test are shown in the arrow diagram:

Functions for which each element of the domain is associated onto a different element of the range are said to be one-toone. Relationships which are one-to-many can occur, but from our preceding definition, they are not functions.

Example 2 

Exercise 1 

Classification of functions

Activity 2

State which of the following functions is a polynomial, rational or irrational function 

a)   Constant function 

A function that assigns the same value to every member of its domain is called a constant function C.

Example 6

Remark:

The constant function that assigns the value c to each real number is sometimes called the constant function c.

Example 7

b) Monomial 

A function of the form cxⁿ , where c is constant and n a nonnegative integer is called a monomial in x.

Example 8

c) Polynomial 

A function that is expressible as the sum of finitely many monomials in x is called polynomial in x.

Example 9

d) Rational function 

A function that is expressible as a ratio of two polynomials is called rational function. It has the form 

Example 10

e) Irrational function 

A function that is expressed as root extractions is called 

Example 11

Finding domain of definition

Activity 3

For which value(s) the following functions are not defined: 

Case 1: The given function is a polynomial

Case 2: The given function is a rational function 

Exercise 2 

Find the domain of definition for each of the following functions: 

Case 3: The given function is an irrational function

Activity 4

For each of the following functions, give a range of values of the variable x for which the function is not defined. 

Example 12

Example 13

Solution

Example 14

Solution

Exercise 3 

Find the domain of definition for each of the following functions; 

Operations on functions

Activity 5 

Just as numbers can be added, subtracted, multiplied and divided to produce other numbers, there is a useful way of adding, subtracting, multiplying and dividing functions to produce other functions. These operations are defined as follows: 

Example 19

Solution

Example 20

Solution

Example 21 

Example 22

Solution

Example 23

Solution

To find the domain of this function, we need to divide it into other functions.  

Exercise 4

Further considerations Odd and even functions

Activity 6 

Even Function 

Example 24

Odd function 

The graph of such a function looks the same when rotated through half a revolution about 0. This is called rotational symmetry.

Example 25

Example 26

Exercise 5 

Study the parity of the following functions: 

Composite functions

Activity 7

Example 27

Solution 

Example 28

Example 29

Solution

Example 30 

Solution

Exercise 6

The inverse of a function 

Activity 8

Find the value of x in function of y if 

Example 31

Solution

Example 32

Find the inverse of the function f (x)=3x–1

Solution

Example 33

Solution

Another method is to write the separate operations of the function as follow chart. We, then, reverse the flow chart, writing the inverse of each operation.

Example 34

Solution

Example 35

Solution

Example 36

Solution

Exercise 7 

Find the inverse of the following functions 

3. Applications

Activity 9

Give three examples of where you think functions can be used in daily life.

Polynomials are used to describe curves of various types; people use them in the real world to graph curves. 

For example, roller coaster designers may use polynomials to describe the curves in their rides. Polynomials can be used to figure how much of a garden’s surface area can be covered with a certain amount of soil. The same method applies to many flat-surface projects, including driveway, sidewalk and patio construction. Polynomials can also be used to model different situations, like in the stock market to see how prices will vary over time. Business people also use polynomials to model markets, as in to see how raising the price of a good will affect its sales. For people who work in industries that deal with physical phenomena or modeling situations for the future, polynomials come in handy every day. These include everyone from engineers to businessmen. For the rest of us, they are less apparent but we still probably use them to predict how changing one factor in our lives may affect another--without even realizing it. 

Functions are important in calculating medicine, building structures (houses, businesses,…), vehicle design, designing games, to build computers (formulas that are used to plug to computer programs), knowing how much change you should receive when making a purchase, driving (amount of gas needed for travel).

Example 37

Solution

Unit summary 

Revision exercise