Types of Functions and their Graphs

Introduction

A function expresses the relationship between an independent variable and a dependent variable. For example, let’s take x and y to be two variables. The value of x is twice that of y, i.e., x=2y. This is a simple linear function of the relationship between x and y. Such a representation is known as a function. To understand this in terms of the set theory, a function relates an element x to an element f(x) in another set. The set of values of x is called the domain of the function. The set of values of f(x) generated by the values in the domain is called the range of the function. It can be represented on a graph as well. Functions and their graphs are the fundamentals of calculus.

 Types of Functions

There are various types of functions formulated to represent functions accurately. Each function looks different on a graph. Here are the main types of functions;

Common Functions

Functions that have more than two variables are known as standard functions. They are also called multivariate functions or multivariable functions—for example, the area of a rectangle= length x breadth. Thus, the area of a rectangle is a function of two variables, i.e., length and breadth. So the function ‘area’ is dependent upon two variables, and its value will change every time the length or breadth changes. It can have a range of real numbers as its value. The area is an example of a standard function.

Polynomial Function and their Graphs

A polynomial function is a function of multiple variables characterized by the highest exponent of an independent variable—for example, the area of a circle=r2. Here, the radius of a circle ‘r’ is an independent variable with an exponential power to the degree 2. Hence, the function(area of a circle) is polynomial. Unique names are commonly used for the functions of powers from one to five, namely linear, quadratic, cubic, quartic, and quintic, for the highest powers being 1, 2, 3, 4, and 5, respectively.

Polynomial functions can be represented on a graph. The independent variable x is plotted along the x-axis, and the dependent variable y is plotted along the y-axis. When the graph of a function of x and y is plotted in the x-y plane, the relation is a function of a vertical line that always passes through only one point of the graphed curve; that is, there would be only one point f(x) corresponding to each x, which is the definition of a function. The function graph then consists of the points with coordinates (x, y) where y = f(x).

Trigonometric Functions

In trigonometric functions, the independent variable always represents an angle. For instance, if A= sin(x), then A is the sin function of an independent variable x. This variable x can be any angle from 0 to 360. Trigonometric functions such as sin x, cos x, tan x, etc., are formulated to show periodic cycles because the range is fixed.

Exponential Functions

An exponential function is a relation in the form y =  ax. Here, x is an independent variable that ranges over the entire real number line as the positive number a. One of the most important of the exponential functions is y =  ex. It can also be written as y = exp (x). Here, e (2.7182818…) has a base of the natural system of logarithms. It is to be noted that x is a logarithm. Thus, a logarithmic function is the inverse of the exponential function. So, if y =ex, then x = ln y. 

Non-algebraic functions, such as exponential and trigonometric, are also considered transcendental functions.

Conclusion

Functions and their graphs have a fundamental role in the world of calculus. In advanced topics such as derivatives, differentiation, and integration, functions are the foundation. The use of operations is so broad that it is difficult to explain it subject-wise. From engineering and architecture to predict the fuel efficiency of a car or implement complex architecture to astronomy and big data computing, functions are used in every field directly and indirectly.