**FUNCTIONS **

**Definitions **

**Variables**

A variable is any quantity that assumes different values in a particular analysis. Examples are:

- Production costs
- Material costs
- Sales revenue

**Constant**

This is any quantity whose value remains unchanged in a particular analysis Examples ar

- Fixed costs
- Rents
- Tuition fees:

**Note:** In a given analysis there are two types of variables namely:

- Independent variable/predictor variable – is that which influences the value of the other variables in a particular analysis.
- Dependent / response variable – is that whose value is influenced or changes when the value of other variables (independent) changes.

**3. Functions **

A function is a mathematical expression which describes a relationship between two or more variables in a particular analysis specifically one dependant variable and one or more independent variables.

**Examples **

If the price of the consumer product is Sh 40 per Kg, then the total sales revenue, S when Q units of the products are produced and sold is obtained as follows:

S = 40q

In this case S is the dependent variable, q the independent variable and 40 is a constant. In terms of number of variables in a function, functions can be classified into the following categories:

- Univariate function
- Bivariate function
- Multivariate function

**A univariate function** is that which involves two variables only, one dependent variable and one independent and is generally written as:

y = f (x) where y = dependent variable

x = independent variable

and f(x) = Function of x

**Example of univariate function **

The price of a house is dependent among other factors, on the size of the house. In functional form, this could be written as follows:

Price = f (size)

Where price is dependent variable Size is independent variable

**A Bivariate function** is that which involves three variables only, one dependent variable and two independent variables:

**Example **

A student’s performance or grade in an examination could be dependent upon the following factors

- IQ
- Time spent on studying in terms of Hours, H

In functional form, this is written as follows:

Grade = f (IQ,H)

Grade is dependent variables

IQ, H Are independent variables

**Multivariable function** is that function which involves four or more variables, one dependent variable and three or more independent variables.

**Example **

The price of a house depends on the following factors:

- Size
- Location
- Security
- Nature of the house

In functional form this is written as follows; Price = f (size, location, security, nature of the house)

Where price – is dependent variable

Size, location, security, nature of the house are independent variables.

**Graph of a function **

A graph is a visual method of illustrating the behaviour of a particular function. It is easy to see from a graph how as x changes, the value of 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 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.

**TYPES OF FUNCTIONS IN BUSINESS **

These include

- Linear functions
- Quadratic functions. Polynominals
- Cubic functions
- Exponential functions
- Logarithmic functions
- Hybrid functions
**Linear functions**

A linear function is a first degree polynomial function that takes the following general form.

y= a +bx

Where y is dependent variable

x is independent variable

a is y-intercept or the value of y when x = 0

b is the slope or gradient or the amount by which y changes in value when x changes by a unit

**Properties/characteristics of linear functions **

When plotted on an x-y coordinate system, the result is a straight line whose general direction is dependent on the slope, b of the function.

Specifically, if