Post 5: Types of Functions

The expression y=f(x) is a general statement to the effect that a mapping is possible between x and y variable. Now let us consider several specific types of function, each representing a rule of mapping.

Constant Function:
A function whose range consists of only one element is called a constant function. As an example, we cite the function:
                                            y = f(x) = 7

In coordinate plane, such a function appear as a horizontal straight line.

Polynomial Function:
The word polynomial means "multiterm", and a polynomial function of a single variable x has the general form:
                                           

Depending on the value of integer n, we have several subclasses of polynomial:
* Constant Function
* Linear Function - As shown below, gives the vertical intercept and gives the slope of the curve. If > 0, the slope of the line would be +ve, else -ve.

                                            

* Quadratic Function - A quadratic function plots as a parabola - a curve with a single built-in bump, as shown below. The curve below implies a +ve ; in the case of < 0, the curve will open the other way, displaying a hill rather than a valley.

* Cubic Function - The graph of a cubic function will, in general, manifest two wiggles, as shown below.



                    

and so forth.

Rational Functions:

A function such as:
                                  
in which, y is expressed as a ratio of 2 polynomials in the variable x, is known as a rational function. A special rational function that has  interesting application in economics is the function:
                                     y = a / x    or   xy = a
which plots as a rectangular hyperbola, as shown below.
                
                                  

This function may be used to represent that special demand curve - with price P and quantity Q on the two axes - for which the total expenditure PQ is constant at all level of price.

The rectangular hyperbola drawn from xy = a never meets the axes, even if it is extended indefinitely upward and to the right. Rather the curve approaches the axes asymptotically: as y becomes large, the curve will come ever closer to the y axis but never actually reach it, and similarly for x axis.

Nonalgebraic Functions:
Exponential functions, such as , in which the independent variable appears in the exponent, are nonalegbraic. The closely related logarithmic functions, such as , are also nonalgebraic. Other types of nonalgebraic functions are trigonometric (or circular) functions.