Special Functions

Introduction:

A function in mathematics is a relationship between a group of inputs with one output. A function, in simple terms, is a relationship between inputs in which each input is associated with only one output. Each function has a domain and a co-domain, often known as a range. The general notation is f(x), where x represents the input. A function’s general representation is y = f(x).

Here, we’ll learn about special functions in mathematics and their various types.

A function expresses the relationship between an independent variable and a dependent variable. To understand this in terms of the set theory, a role 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.

Types of Special 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:

Polynomial Function

A polynomial function is a function of multiple variables characterised 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

Trigonometric functions were first introduced by Indian mathematicians in 432 AD. Although, they didn’t understand trigonometry as we do today. It wasn’t really developed until the 16th century. This was when European mathematicians started using algebra to solve trigonometric equations. 

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

A basic exponential function is of the form f(x)=ax, where ‘a’ is a constant and x is a variable, according to its definition. f(x)=ex, where ‘e’ is “Euler’s number” and e = 2.718, is a widely used exponential function. An exponential function may include a constant that is a multiple of the variable in its power if the possibilities of distinct exponential functions are extended. f(x)=ekx, for example, is another form of exponential function. It can also have the form f(x)=pekx, with ‘p’ being a constant. As a result, an exponential function can take any of the following shapes:

• f(x)=ax
• f(x)=bax
• f(x)=baxy
• f(x)=ex
• f(x)=ekx
• f(x)=pekx

Here a>0 and a1

All other letters are constants except ‘x,’ ‘x’ is a variable, and f(x) is an exponential function in terms of x. Also, every exponential function must have a positive base. Specifically, a > 0 and e > 0 in the above functions. Also, a should not equal 1 (if a = 1, the function f(x)=ax becomesf(x)=1, and the function is linear but not exponential in this case).

Logarithmic Function 

The logarithmic function is the inverse of exponentiation in mathematics. The logarithmic function definition is as follows:

For x > 0 , a > 0, and a (subscript) ≠ 1,

y = logax if and only if x = a^y

Then, the function is:

f(x) = logax

The logarithm’s base is equal to a. This is written as log base a of x. Base 10 and base e are the most commonly used bases in logarithmic functions.

Typically, there are two types of logarithmic functions:

• A popular logarithmic function’s base is 10. The log 10 or log function is used to represent this function.
• The natural logarithmic function’s base is e. The letters ln or loge stand for this function.

Inverse Trigonometric Functions

• ƒ: [−π/2, π/2]  ⇒ [-1, 1]  is defined as ƒ(x) = sin(x) and is a bijection, hence the inverse exists. The inverse of sin-1  is also called arcsine, and inverse functions are also called arc functions.
• ƒ:[−π/2 , π/2] ⇒ [−1 , 1] is defined as sinθ = x ⇔ sin-1(x) = θ , θ belongs to [−π/2 , π/2] and x belongs to [−1 , 1]  .

Implicit Function

A function developed for differentiation of functions comprising variables but not easily written in y = f(x)  is called an implicit function (x). If x2 + y2 + 4xy + 25 = 0, then the dependent and independent variables “y” cannot be easily separated to describe it as a function of “y = f(x),” then it is an example of an implicit function (x). 

Properties of implicit functions: 

• It is impossible to write y = f(x) for the implicit function f(x). 
• When the function is implicit, it is always interpreted as f(x,y) = 0. 
• There are numerous variables involved in the implicit function. 
• The dependent and independent variables are used to formulate the implicit function. 
• Several points are crossed by the vertical line drawn along the graph of an implicit function.

Conclusion :

In this article, we have learned various types of special functions, like  Polynomial Function,  Trigonometric Functions, Exponential Functions, Logarithmic, Implicit Function, Inverse Trigonometric Functions, their definition, formulae, properties, rules, uses, and multiple examples also. This topic has a vast scope, and the numerical question asked from this topic has an easy to moderate level. So, read the full notes to clear all the concepts of this topic.

This brief note on special functions can help you learn and grip numerical functions more efficiently.