FUNCTIONS AND EQUATIONS

Introduction
The study of functions and equations is important and thus popular in the business and
economic world because the basic understanding of concepts of functions and equations
provides a form of logical language through which business specialists can communicate
important concepts and ideas.

Objectives
Objectives by the end of this topic you should be able to:
i. Define functions and equations
ii. Describe the various types of functions
iii. Understand the various methods of solving quadratic functions
iv. Sketch the graphs of linear and quadratic functions
v. Apply linear and quadratic functions in solving business problems
Learning activities
Learning Activity 1.1: Reading
Read the provided topic notes on functions and equations. You have also been provided some
links to the mathematics for business resources.
Learning Activity 1.2: topic questions
Attempt the topic questions at the end of the topic
Assessment
Activity 1.2 on the topic questions will be graded
Topic Resources
Attached topic 1 notes
URL Links
https://www.researchgate.net/publication/281838644_An_Introduction_to_Business_Mathematics/
download

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 Topic 1 Notes
Functions and graphs
A function is a mathematical relationship in which the value of a single dependent variable is
determined from the values of one or more independent variables. The following is an
example of a function in which y is said to be a function of x.

y = a + bx

In the above example, both x and y are variables this is because they may assume
different values throughout the analysis of the function. On the other hand, a and b are
referred to as constants because they assume fixed values.

The variable y is a dependent variable in the sense that its values are generated from an
independent variable x.

The collection of all the values of the independent variable for which the function is defined
is referred to as the domain of the function corresponding to this, we have the range of the
function, which is the collection of all the values of the dependent variable defined by the
function

The fact that it is a function of x can also be denoted by the following general
form y = f(x)
Functions of a single independent variable may either be linear or
nonlinear. Linear functions can be represented by:
y = a + bx

Whereas non – linear functions can be represented by functions such as:

3 3
i y = α0 + α 1x + α2x
2
ii. y = 3x + 18
2
iii. y = 2x + 5x + 7
2
iv. ax + bx + cy + d = 0 v. xy = k
x
vi. y=a

Where α, a, b, c, d, k = constants

Graph of a function

A graph is a visual method of illustrating the behavior of a particular function. It is easy to see
from a graph how as x changes, the value of the f(x) is changing.

The graph is thus much easier to understand and interpret than a table of values. For
example, by looking at a graph we can tell whether f(x) is increasing or decreasing as x
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 increases or decreases.

We can also tell whether the rate of change is slow or fast. Maximum and minimum values
of the function can be seen at a glance. For particular values of x, it is easy to read the
values of f(x) and vice versa i.e. graphs can be used for estimation purposes

Different functions create different shaped graphs and it is useful knowing the shapes of
some of the most commonly encountered functions. Various types of equations such as
linear, quadratic, trigonometric, exponential equations can be solved using graphical
methods.

Equat
ions

An equation is an expression with an equal sign

(=)
Equations are classified into two main groups linear equations and nonlinear equations.
Examples of linear equations are

x + 13 = 15

7x + 6 = 0

Nonlinear equations in the variable x are equations in which x appears in the second or
higher degrees. They include quadratic and cubic equations amongst others. For example
2
5x + 3x + 7 = 0 (quadratic equation)

3 2
2x + 4x + 3x + 8 = 0 (cubic equation)

The solution of equations or the values of the variables for which the equations hold is called
the roots of the equation or the solution set.

Solution of Linear Equation

Supposing M, N, and P are expressions that may or may not involve variables, then the
following constitute some rules which will be useful in the solution of linear equations

Rule 1: Additional rule

If M = N then M + P = N + P

Rule 2: Subtraction rule

If M = N, Then M – P = N – P

Rule 3: multiplication rule

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 If M = N and P ≠ O then M x P = N x P

Rule 4: Division rule

If P x M = N and P ≠ O

And N/P = Q, Q being an arterial number then

M = N/P

Example

i. Solve 3x + 4 = - 8

ii. Solve
y=
-4
3
Solutions

i. 3x + 4 = –8
3x + 4 – 4 = – 8 – 4 ( by subtraction rule)
3x = – 12 ( simplifying)

3x
12



3 3 (by division rule)

x=–4 ( simplifying)
y
3  4  3
3

y = –12 ( simplifying)

Solution of quadratic equations

Suppose that we have an equation given as

2
follows ax + bx + c = 0
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 Where a, b and c are constants, and a≠ 0. such an equation is referred to as the general
quadratic
equation in x. if b = 0, then we have

2
ax + c = 0
Which is a pure quadratic equation

There are 3 general methods for solving quadratic equations; solution by factorization,
solution by completing the square and solution by the quadratic formula.
Solution by Factorization
The f o l l o w i n g a r e the general steps commonly used in solving quadratic equations by
factorization
Example
(i) Set the given quadratic equation to zero
(ii) Transform it into the product of two linear factors
(iii)Set each of the two linear factors equal to zero (iv)Find
the roots of the resulting two linear equations
Solve the following equation by
2
factorization i. 6x = 18x
2
ii. 15x + 16x = 15
Solutions
2
i. 6x = 18x
2
6x – 18x = 0 ............................................(step 1)

6x (x – 3) = 0.............................................(step 2)

6x = 0 .......................................................(step 3)

and x – 3
=0

∴ x = 0 or x = 3 ......................................(by step 4)

2
ii. 15x + 16x = 15
2
15x + 16x – 15 = 0 .................................. (step 1)
 (5x – 3) (3x +5) = 0..................................(step 2)
(5x–3 =0} (Step 3
{3x + 5 = 0}

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 ∴x=-5
or + 3 ................................. .(step 4) 
3 5
Solution by Completing the Square

The process of completing the square involves the construction of a perfect square from
the members of the equation which contains the variable of the equation.
2
Consider the equation – 9x – bx = 0
The method of completing the square will involve the following steps

2
i.  Make the coefficient of x unity
ii. 
Add the square of ½ the coefficient of x to both sides of the equal sign.
The left-hand side is now a perfect square
iii.  Factorize the perfect square on the left-hand
side. iv.  Find the square root of both sides
v.  Solve for x

Example

Solve by completing the square.

2
i. 3x = 9x
2
ii. 2x + 3x + 1 = 0

Solutions
i. 3x2 = 9x or
2
(3x - 9x = 0)
x2 - 3x = 0............................................................. (Step 1)

( ) ( )
2 2
2 −3 −3
x −3 x+ = .....................................(Step 2)
2 2

( )
2
3 9
x− = .........................................................(Step 3)
2 4

x−3=±
√ 9
4
........................................................(Step 4)

3+3 3 3
¿ ∨ − =¿
2 2 2
(= 3 or 0)

ii. 2x2 + 3x + 1 = 0 or (2x2 + 3x = -1)

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 2 3x 1
x + = - …………………...….. (Step 1)
2 2

() ()
2 2
23x 3 3 1
X + + = − ……… (Step 2)
2 4 4 2

( )
2
3 1
x+ = …………………….. (Step 3)
4 16

x+
3
4
=±
√ 1
16
3 1
x=− =±
4 4
−3 1 3 1
+ or - -
4 4 4 4
−1
x= or x=−1
2

Solution by Quadratic Formula

Consider the general quadratic equation
2
a x + bx + c = 0 where a ≠0
The roots of the equation are obtained by the following formula:
−b ± √ b2−4 ac
x=
2a
Example
Solve for x by formula
5x2 + 2x – 3 = 0
Solution
a = 5, b = 2, c = - 3
−b ± √ b2−4 ac
x=
2a

−2 ± √ 22−4(5)(−3)
x=
2(5)

3
x= ∨−1
5

Inequalities
An inequality or inequation is an expression involving an inequality sign (i.e. >, <, ≤, ≥, i.e.
greater than, less than, less or equal to, greater or equal to) The following are some examples
of inequations in variable x.
3x + 3 > 5
x2 – 2x – 12 < 0

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The first is an example of linear inequation and the second is an example of a quadratic in
equation.
Solutions of inequations
The solutions set of inequations frequently contain many elements. In a number of cases they
contain infinite elements.

Example
Solve and graph the following inequation
x – 2 > 2; x ⊂ w (where x is a subset of w)

Solution
x – 2 > 2 so x – 2 + 2 > 2 + 2
Thus, x>4
The solution set is infinite, being all the elements in w greater than 4

0 1 2 3 4 5 6 7 8 9 10 11

Example
Solve and graph
3x – 7 < - 13;
Solution
3 x−7<−13
⇒ 3 x−7+ 7<−13+7
⇒ 3 x <−6
3 x −6
<
3 3
-4 -3 -2 -1 0 2 x <−2 3 4

….. R Line

Rules for solving linear inequations

Suppose M, M1, N, N1 and P are expressions that may or may not involve variables, then the
corresponding rules for solving inequations will be:
Rule 1: Addition rule
If M > N and M1> N1
Then M + P > N + P and
M1 + P >N1+ P
Rule 2: Subtraction Rule
If M < N and M1 ≥N1
Then M – P < N – P and
M1 – P ≥N1– P
Rule 3: Multiplication rule
If M ≥N and M1 > N1 and P≠ 0
Then MP ≥NP; M1P > N1P
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 M(-P) ≤ N(-P) and M1(-P) < N1(-P)
Rule 4: Division
If M > N and M1< N1 and P≠ 0
Then M/P > N/P: M1/P < N1/P
M/(-P) < N/(-P): and M1/(-P) > N1/(-P)
Rule 5: Inversion Rule
If M/P ≤ N/Q where P, Q ≠ 0
M1/P > N1/Q
Then P/M ≥ Q/N and P/M1 < Q/N1
Note: The rules for solving equations are the same as those for solving equations with one
exception; when both sides of an equation are multiplied or divided by a negative number, the
inequality symbol must be reversed (see rule 3 & Rule 4 above).

Example
Solve and graph the following:
i. 7 – 2x > - 11 ;
ii. –5x + 4 ≤ 2x – 10 ;
iii. –3 ≤ 2x + 1 < 7;
Solutions

line Q
-3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 11

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 Q
line
-4 -3 -2 -1 0 1 2 3 4 5

Q
line
-4 -3 -2 -1 0 1 2 3 4 5

Linear inequation in two variables: relations

An expression of the form
y ≥ 2x – 1
is technically called a relation. It corresponds to a function, but different from it in that, corresponding
to each value of the independent variable x, there is more than one value of the dependent variable y
Relations can be successfully presented graphically and are of major importance in linear
programming.

1.2 Linear simultaneous equations:

Two or more equations will form a system of linear simultaneous equations if such equations
be linear in the same two or more variable

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For instance, the following systems of the two equations is simultaneous in the two variables
x and y.
2x + 6y = 23
4x + 7y = 10
The solution of a system of linear simultaneous equations is a set of values of the variables
which simultaneously satisfy all the equations of the system.

Solution techniques
a) The graphical technique
The graphical technique of solving a system of linear equations consists of drawing the
graphs of the equations of the system on the same rectangular coordinate system. The
coordinates of the point of intersection of the equations of the system would then be the
solution.

Example
10

9
.

8
.
7
.
6
. (2,4)
5
.
4
.
3 x + 2y = 10
.
2x + y = 8
2
.

-11 1 2 3 4 5 6 7 8 9 10 11 12 13

The above figure illustrates:

Solution by graphical method of two equations
2x + y = 8
x + 2y = 10
The system has a unique solution (2, 4) represented by the point of intersection of the two
equations.

b) The elimination technique

This method requires that each variable be eliminated in turn by making the absolute value of
its coefficients equal in the equations of the system and then adding or subtracting the
equations. Making the absolute values of the coefficients equal necessitates the multiplication
of each equation by an appropriate numerical factor.
Consider the system of two equations (i) and (ii) below
2x – 3y = 8 …….. .........................................(i).
3x + 4y = -5 ……...........................................(ii).
Step 1
Multiply (i) by 3

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6x – 9y = 24 …… .....................................................(iii).
Multiply (ii) By 2
6x + 8y = - 10 …… ...................................................(iv).
Subtract iii from iv.
17y = -34 ……...........................................................(v).
∴ y = -2

Step 2
Multiply (i) by 4
8x – 12y = 32 ……. ..................................................(vi)
Multiply (ii) by 3
9x + 12y = -15 ….. ....................................................(vii)
Add vi to vii
17x = 17 ……............................................................(viii)
∴ x=1
Thus x = 1, y = -2 i.e. {1,-2}

c) The substitution technique

To illustrate this technique, consider the system of two equations (i). and (ii) reproduced
below
......2x – 3y = 8 ……... (i).
......3x + 4y = -5 …… (ii).
The solution of this system can be obtained by
a) Solving one of the equations for one variable in terms of the other variable;
b) Substituting this value into the other equation(s) thereby obtaining an equation with
one unknown only
c) Solving this equation for its single variable finally
d) Substituting this value into any one of the two original equations so as to obtain the
value of the second variable

Step 1
Solve equation (i) for variable x in terms of y
2x – 3y = 8
x= 4 + 3/2 y (iii)
Step 2
Substitute this value of x into equation (ii). And obtain an equation in y only
3x + 4y = -5
3 (4 + 3/2 y) + 4y = -5
8 ½ y = - 17 ……. (iv)
Step 3
Solve the equation (iv). For y
8½y = -17
y = -2
Step 4
Substitute this value of y into equation (i) or (iii) and obtain the value of x
2x – 3y = 8
2x – 3(-2) = 8
x=1
Example
Solve the following by substitution method

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 2x + y = 8
3x – 2y = -2
Solution
Solve the first equation for y
y = 8 – 2x
Substitute this value of y into the second equation and solve for x
3x – 2y = -2
3x – 2 (8-2x) = -2
x=2
Substitute this value of x into either the first or the second original equation and solve for y
2x + y = 8
(2) (2) + y = 8
y=4
Topic questions
(a) Solve the following simultaneous equations by elimination method
(i) 2x – 3y = 8
3x + 4y = -5

(ii) Solve the following simultaneously.

X=y-4
3 3
2x – 5y = -7
b) Solve the following quadratic by factoring
4x2+8x+3=0
References
A Asano (2013) An Introduction to Mathematics for Economics (Cambridge: Cambridge
University Press) ISBN–13: 9780521189460
Budnick F.S 2001: Applied mathematics for business, economic and social sciences.
E T Dowling (2009) Schaum’s Outline of Mathematical Methods for Business and
Economics
(New York: McGraw–Hill) ISBN–13: 9780071635325
E T Dowling (1990) Schaum’s Outline of Calculus for Business, Economics, and the Social
Sciences (New York: McGraw–Hill) ISBN–13: 9780070176737
Larson, R. E. and Hostetlerk, R. P., 1993. Precalculus – A Graphic Approach; D. C Health
and Company,
Margaret, L. L., Hornsby, E. J. and Miller, C. D. 1995. Introductory Algebra; Harper
Collins
College Publishers,

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