Business Mathematics MTH367 Lecture 4 Chapter 4 Mathematical
![Business Mathematics MTH-367 Lecture 4 Business Mathematics MTH-367 Lecture 4](https://slidetodoc.com/presentation_image_h2/5f78beec94e73350f6332c8b95c9e9c9/image-1.jpg)
Business Mathematics MTH-367 Lecture 4
![Chapter 4 Mathematical Functions Chapter 4 Mathematical Functions](http://slidetodoc.com/presentation_image_h2/5f78beec94e73350f6332c8b95c9e9c9/image-2.jpg)
Chapter 4 Mathematical Functions
![Objectives • Enable the reader to understand the nature and notation of mathematical functions Objectives • Enable the reader to understand the nature and notation of mathematical functions](http://slidetodoc.com/presentation_image_h2/5f78beec94e73350f6332c8b95c9e9c9/image-3.jpg)
Objectives • Enable the reader to understand the nature and notation of mathematical functions • Provide illustrations of the application of mathematical functions • Provide a brief overview of important types of functions and their characteristics • Discuss the graphical representation of functions
![Review Ø Solution of system of linear equations with n> 2. Ø Application to Review Ø Solution of system of linear equations with n> 2. Ø Application to](http://slidetodoc.com/presentation_image_h2/5f78beec94e73350f6332c8b95c9e9c9/image-4.jpg)
Review Ø Solution of system of linear equations with n> 2. Ø Application to product mix problem
![Today’s Topics • • Functions Domain and Range of a function Multivariate functions Types Today’s Topics • • Functions Domain and Range of a function Multivariate functions Types](http://slidetodoc.com/presentation_image_h2/5f78beec94e73350f6332c8b95c9e9c9/image-5.jpg)
Today’s Topics • • Functions Domain and Range of a function Multivariate functions Types of functions
![Why functions? • Identification of the relevant mathematical representation of the real world phenomenon Why functions? • Identification of the relevant mathematical representation of the real world phenomenon](http://slidetodoc.com/presentation_image_h2/5f78beec94e73350f6332c8b95c9e9c9/image-6.jpg)
Why functions? • Identification of the relevant mathematical representation of the real world phenomenon is done by mathematical modelling. • If a model is a good approximation, it can be very useful in studying the reality and decision making in real world problems. • In mathematical models, the significant relationships are typically represented by mathematical functions.
![Intuition of function • Intuition of function •](http://slidetodoc.com/presentation_image_h2/5f78beec94e73350f6332c8b95c9e9c9/image-7.jpg)
Intuition of function •
![Function • Definition: A function is a mathematical rule that assigns to each input Function • Definition: A function is a mathematical rule that assigns to each input](http://slidetodoc.com/presentation_image_h2/5f78beec94e73350f6332c8b95c9e9c9/image-8.jpg)
Function • Definition: A function is a mathematical rule that assigns to each input value one and only one output value. • Definition: The domain of a function is the set consisting of all possible input values. • Definition: The range of a function is the set of all possible output values.
![Some examples • The fare of taxi depend upon the distance and the day Some examples • The fare of taxi depend upon the distance and the day](http://slidetodoc.com/presentation_image_h2/5f78beec94e73350f6332c8b95c9e9c9/image-9.jpg)
Some examples • The fare of taxi depend upon the distance and the day of the week. • The fee structure depend upon the program and the type of education (on campus/off campus) you are admitting in. • The house prices depends on the location of the house.
![Notation • The assigning of output values to corresponding input values is often called Notation • The assigning of output values to corresponding input values is often called](http://slidetodoc.com/presentation_image_h2/5f78beec94e73350f6332c8b95c9e9c9/image-10.jpg)
Notation • The assigning of output values to corresponding input values is often called as mapping. The notation represents the mapping of the set of input values of x into the set of output values y, using the mapping rule.
![Notation (contd. ) • Notation (contd. ) •](http://slidetodoc.com/presentation_image_h2/5f78beec94e73350f6332c8b95c9e9c9/image-11.jpg)
Notation (contd. ) •
![• The input variable is called the independent variable and the output variable • The input variable is called the independent variable and the output variable](http://slidetodoc.com/presentation_image_h2/5f78beec94e73350f6332c8b95c9e9c9/image-12.jpg)
• The input variable is called the independent variable and the output variable is called the dependent variable. • Note that x is not always the independent variable, y is not always the dependent variable and f is not always the rule relating x and y. • Once the notation of function is clear then, from the given notation, we can easily identify the input variable, output variable and the rule relating them, e. g. u=g(v) v = input variable, u= output variable g= rule relating u and v
![Example • Example •](http://slidetodoc.com/presentation_image_h2/5f78beec94e73350f6332c8b95c9e9c9/image-13.jpg)
Example •
![Example-Weekly Salary Function • A person gets a job as a salesperson. • His Example-Weekly Salary Function • A person gets a job as a salesperson. • His](http://slidetodoc.com/presentation_image_h2/5f78beec94e73350f6332c8b95c9e9c9/image-14.jpg)
Example-Weekly Salary Function • A person gets a job as a salesperson. • His salary depends upon the number of units he sells each week. Then, dependency of weekly salary on the units sold per week can be represented as: where f is the name of the salary function.
![Example (contd. ) • Example (contd. ) •](http://slidetodoc.com/presentation_image_h2/5f78beec94e73350f6332c8b95c9e9c9/image-15.jpg)
Example (contd. ) •
![Example • Example •](http://slidetodoc.com/presentation_image_h2/5f78beec94e73350f6332c8b95c9e9c9/image-16.jpg)
Example •
![Ways to define a function • In words: The output function is square of Ways to define a function • In words: The output function is square of](http://slidetodoc.com/presentation_image_h2/5f78beec94e73350f6332c8b95c9e9c9/image-17.jpg)
Ways to define a function • In words: The output function is square of the input function plus 1 This method is not easy and practical when functions involve more variables or more terms.
![Domain and Range • Recall that the set of all possible input values is Domain and Range • Recall that the set of all possible input values is](http://slidetodoc.com/presentation_image_h2/5f78beec94e73350f6332c8b95c9e9c9/image-18.jpg)
Domain and Range • Recall that the set of all possible input values is called the domain of a function. Domain consists of all real values of the independent variable for which the dependent variable is defined and real. • To find the domain of a function, we look at few examples.
![Examples • Examples •](http://slidetodoc.com/presentation_image_h2/5f78beec94e73350f6332c8b95c9e9c9/image-19.jpg)
Examples •
![Examples • Examples •](http://slidetodoc.com/presentation_image_h2/5f78beec94e73350f6332c8b95c9e9c9/image-20.jpg)
Examples •
![Examples • Examples •](http://slidetodoc.com/presentation_image_h2/5f78beec94e73350f6332c8b95c9e9c9/image-21.jpg)
Examples •
![Examples • Examples •](http://slidetodoc.com/presentation_image_h2/5f78beec94e73350f6332c8b95c9e9c9/image-22.jpg)
Examples •
![Restricted domain and range • Restricted domain and range •](http://slidetodoc.com/presentation_image_h2/5f78beec94e73350f6332c8b95c9e9c9/image-23.jpg)
Restricted domain and range •
![](http://slidetodoc.com/presentation_image_h2/5f78beec94e73350f6332c8b95c9e9c9/image-24.jpg)
![Multivariate Functions • For many mathematical functions, the value of the dependent variable depends Multivariate Functions • For many mathematical functions, the value of the dependent variable depends](http://slidetodoc.com/presentation_image_h2/5f78beec94e73350f6332c8b95c9e9c9/image-25.jpg)
Multivariate Functions • For many mathematical functions, the value of the dependent variable depends upon more than one independent variable. • Functions which contain more than one independent variables are called multivariate functions. • A function having two independent variables is called bivariate function.
![](http://slidetodoc.com/presentation_image_h2/5f78beec94e73350f6332c8b95c9e9c9/image-26.jpg)
![Types of Functions • • • Constant function Linear function Quadratic function Cubic function Types of Functions • • • Constant function Linear function Quadratic function Cubic function](http://slidetodoc.com/presentation_image_h2/5f78beec94e73350f6332c8b95c9e9c9/image-27.jpg)
Types of Functions • • • Constant function Linear function Quadratic function Cubic function Polynomial function Rational function Combination of functions Composite function Demand function
![Constant Functions Definition: A constant function has the general form Here, domain is the Constant Functions Definition: A constant function has the general form Here, domain is the](http://slidetodoc.com/presentation_image_h2/5f78beec94e73350f6332c8b95c9e9c9/image-28.jpg)
Constant Functions Definition: A constant function has the general form 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 • Marginal Revenue: This revenue is the additional revenue derived from selling one](http://slidetodoc.com/presentation_image_h2/5f78beec94e73350f6332c8b95c9e9c9/image-29.jpg)
• 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 (slopeintercept) form where e, g Linear Functions Definition: A linear function has the general (slopeintercept) form where e, g](http://slidetodoc.com/presentation_image_h2/5f78beec94e73350f6332c8b95c9e9c9/image-30.jpg)
Linear Functions Definition: A linear function has the general (slopeintercept) form where e, g This function is represented by a straight line with slope and y-intercept • The weekly salary function is also an example of linear function.
![Quadratic Functions • Definition A quadratic function has the general form provided that e, Quadratic Functions • Definition A quadratic function has the general form provided that e,](http://slidetodoc.com/presentation_image_h2/5f78beec94e73350f6332c8b95c9e9c9/image-31.jpg)
Quadratic Functions • Definition A quadratic function has the general form provided that e, g
![Cubic Functions • Definition A cubic function has the general form provided that e, Cubic Functions • Definition A cubic function has the general form provided that e,](http://slidetodoc.com/presentation_image_h2/5f78beec94e73350f6332c8b95c9e9c9/image-32.jpg)
Cubic Functions • Definition A cubic function has the general form provided that e, g
![Polynomial Functions • Polynomial Functions •](http://slidetodoc.com/presentation_image_h2/5f78beec94e73350f6332c8b95c9e9c9/image-33.jpg)
Polynomial Functions •
![Rational Functions • Definition: A rational function has the general form Where and functions, Rational Functions • Definition: A rational function has the general form Where and functions,](http://slidetodoc.com/presentation_image_h2/5f78beec94e73350f6332c8b95c9e9c9/image-34.jpg)
Rational Functions • Definition: A rational function has the general form Where and functions, e. g. are both polynomial
![Combination of Functions can be combined algebraically to form a resultant function. If Then Combination of Functions can be combined algebraically to form a resultant function. If Then](http://slidetodoc.com/presentation_image_h2/5f78beec94e73350f6332c8b95c9e9c9/image-35.jpg)
Combination of Functions can be combined algebraically to form a resultant function. If Then these functions can be combined in four different algebraic ways. 1)Sum function:
![Combination of Functions 2) Difference function: 3) Product function: 4) Quotient function: Combination of Functions 2) Difference function: 3) Product function: 4) Quotient function:](http://slidetodoc.com/presentation_image_h2/5f78beec94e73350f6332c8b95c9e9c9/image-36.jpg)
Combination of Functions 2) Difference function: 3) Product function: 4) Quotient function:
![](http://slidetodoc.com/presentation_image_h2/5f78beec94e73350f6332c8b95c9e9c9/image-37.jpg)
![Composite functions • Composite functions •](http://slidetodoc.com/presentation_image_h2/5f78beec94e73350f6332c8b95c9e9c9/image-38.jpg)
Composite functions •
![Example Example](http://slidetodoc.com/presentation_image_h2/5f78beec94e73350f6332c8b95c9e9c9/image-39.jpg)
Example
![Demand functions • Demand functions •](http://slidetodoc.com/presentation_image_h2/5f78beec94e73350f6332c8b95c9e9c9/image-40.jpg)
Demand functions •
![Review • • • Functions Ways to define a function Domain and range of Review • • • Functions Ways to define a function Domain and range of](http://slidetodoc.com/presentation_image_h2/5f78beec94e73350f6332c8b95c9e9c9/image-41.jpg)
Review • • • Functions Ways to define a function Domain and range of a function Multivariate functions Types of functions
![](http://slidetodoc.com/presentation_image_h2/5f78beec94e73350f6332c8b95c9e9c9/image-42.jpg)
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