MTH108 Business Math I

Lecture 9

Chapter 4

Mathematical functions

Review

• Functions• Ways to express a function;• In words; a mapping diagram; using a formula;

by an equation.• Domain and range of a function• Methods to find the domain• Multivariate functions

Today’s Topic

• Types of Functions• Constant functions• Linear functions• Quadratic functions• Cubic functions• Polynomial functions• Rational functions• Combination of functions• Composition of functions• Graphical representation of functions• Vertical line test

Constant Functions

• Functions can be classified according to their structural characteristics.

Definition:A constant function has the form

e.g. Here, domain is the set of all real numbers, i.e.

And range is the single value

• Marginal Revenue: This revenue is the additional revenue derived from selling one more unit of a product on service.

If each unit of a product sells at the same price, the marginal revenue is always equal to the price.

e.g. if a product is sold for 80 rupee per unit, the marginal revenue function can be stated as the constant function.

MR=f(x)=80

Linear Functions

Definition: A linear function has the general (slope-intercept) form

where This function is represented by a straight line with slope

and y-intercept

• Examples

The weekly salary function

Quadratic Functions

• Definition A quadratic function has the general form

Where and are real constants with

Cubic Functions

• Definition A cubic function has the general form

Where and are real constants with

Polynomial Functions

• Definition A polynomial function of degree n has the general form

Where and are real constants with

All the previous types of functions are polynomial functions.

Rational Functions

• Definition A rational function has the general form

Where and are both polynomial functions

Combination of Functions

Functions can be combined algebraically to form a resultant function. If

Then these functions can be combined in four different algebraic ways.

Sum function:Difference function:Product function: Quotient function:

The domain and the range of these functions is the set of values for the independent variables for which both the functions are defined.

Composite functions

• Definition: A composite function exists when one function can be viewed as a function of the values of another function. (when range of one function is the subset of the domain of the other function).

• If

• Consider the weekly salary equation

• If the analysis shows that the quantity sold each week by a salesperson is dependent upon the price charged for the product, This function h is given by

• Then,

Graphical Representation of Functions

• The functions of one independent variable are graphed in two dimensions, 2-space.

• The functions in two independent variables are graphed in three dimension, 3-space.

• For the case of more than two independent variables, graphical representation is lost.

• In case of two variables, rectangular coordinate axes are used. The vertical axis denotes the dependent variable and the horizontal axis denotes the independent variable.

Examples

• Linear functions

Examples

• Quadratic functions

Examples

• Cubic functions

Examples

Vertical Line Test

If a vertical line is drawn through any value in the domain, it will intersect the graph of the function at one point only.

If the vertical line intersects at more than one point then, the graph depicts a relation and not a function.

Review

• Types of functions• Constant; Linear; quadratic; cubic; polynomial;

rational; algebraic combination; composition• Graphical representation of functions• Vertical line test