1. MTH108 Business Math I
Lecture 9

2. Mathematical functions
Chapter 4
Mathematical functions

3. 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

4. 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

5. 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

6. 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

8. 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

9. Examples
The weekly salary function

10. Quadratic Functions
Definition A quadratic function has the general form
Where and are real constants with

11. Cubic Functions Definition A cubic function has the general form
Where and are real constants with

12. 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.

13. Rational Functions Definition A rational function has the general form
Where and are both polynomial functions

14. 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:

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

16. 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

17. 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,

20. 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.

21. Examples
Linear functions

22. Examples
Quadratic functions

23. Examples
Cubic functions

24. Examples

25. 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.

28. Review Types of functions
Constant; Linear; quadratic; cubic; polynomial; rational; algebraic combination; composition
Graphical representation of functions
Vertical line test