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Types of Functions

The mathematical concept of a function expresses the intuitive idea that one quantity completely determines another quantity. An equation will be a function if for any x in the domain of the equation, there is exactly one value of y. More formally, a function is defined a type of relation which has only one output value with respect any permissible input value. The argument and the value may be real numbers, but they can also be elements from any given sets: the domain, the co-domain of the function and all the assigned values are in the Range of the function. It is important how we assign the values to each element of the domain.

There may be different types of functions in mathematics. The classification of functions on the basis of properties may be as follows :
(1) Even Function
(2) Odd Function
(3) Monotonic Function
(4) Surjective Function
(5) Bijective Function
(6) Injective Function

The functions can be classified into following on the basis of variables used :
(1) Algebraic Function
(2) Exponential Function
(3) Logarithmic Function
(4) Analytic Function
(5) Inverse Function
(6) Monotonic Function
(7) Polynomial Function
(8) Linear Function
(9) Quadratic Function
(10) Elliptic Function
(12) Identity Function
(13) Constant Function
(14) Power Function
(15) Vector-valued Function
(16) Integer-valued Function
(17) Periodic Function
(18) Piece-wise linear Function etc.

In this article, we shall learn about few of the above mentioned types of functions.

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Types of Polynomial Functions

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A polynomial or algebraic function is a function which can be expressed in the following form:
$f(x) = a_{n}x^{n} + a_{n - 1}x^{n - 1}+ ..... + a_{2}x^{2} + a_{1}x^{1} + a_{0}$

In this expression, the highest power which is present in the expression is called the degree of the polynomial function. According to the degree, we have some different types of polynomial functions as follows:

  1. If the degree is one then we can say that the given polynomial function is linear.
  2. If the degree is two then we can say that the given polynomial function is quadratic.
  3. If the degree is three then we can say that the given polynomial function is cubic.

Constant Polynomial Functions:

It is a polynomial of the oth degree, where f(x) = cx0 = c(1) = c.It disregards the input and the result is always c. Its graph is a horizontal line.

For example f(x) = 2, whatever the value of x result is always 2.

Linear Polynomial Functions:

It is a polynomial of the first degree, the input should be multiplied by m and it adds to c. It is represented as f(x) = m x + c such as f(x) = 2x + 1 at x = 1.
f(1) = 2 . 1 + 1 = 3
f(1) = 3

Quadratic Polynomial Functions

A polynomial of the second degree is represented as
f(x) = ax2 + bx + c, a $\neq$ 0.

where a, b, c are constant and x is a variable.
For example, value of f(x) = 2x2 + x - 1 at x = 2.
Put x = 2, f(2) = 2.22 + 2 - 1 = 9

Cubic Polynomial Function:

A polynomial of the degree three is called the cubic polynomial and is denoted as follows:
f(x) = ax3 + bx2 + cx + d, a $\neq$ 0.

Types of Functions Graphs

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Given below are the graphs of different types of functions.

Odd and Even Functions

When there is no change in the sign of f(x) when x is changed to –x, then that function is called an even function. (i.e) f(-x) = f(x)

Let us consider $f(x) = 2x^{2} - 3$
$f(-x) = 2(-x)^{2} - 3 = 2x^{2} - 3$
$f(x) = f(-x)$

Other examples are, $g(x) = \cos x$ , $h(x) = e^{x} + e^{-x}$

The graph of an even function is such that the two ends of the graph will be directed towards the same side. We can observe this from the following graph of $y = 2x^{2} - 3$. The graph remain unaltered when reflected about the y-axis.

Types of Functions

When the sign of f(x) is changed when x is changed to –x, then it is called an odd function.
(i.e) f(-x) = - f(x).

Let us consider $f(x) = x^{3} + 6x$
$f(-x) = (-x)^{3} + 6(-x) = -x^{3} - 6x = -(x^{3} + 6x) = -f(x)$
$f(x)$ is an odd function.

Other examples are $g(x) = \tan x$ , $h(x) = x^{5} - 3x^{3} + 9x + 8$
The following graph shows the odd function, $f(x) = x^{3}$ , and its reflection about the y-axis, which is $f(-x) = -x^{3}$.

Types of Function

Inverse Functions

From every function $y = f(x)$, we may be able to deduce a function $x = g(y)$, such that the composition $fog = Iy$ , $gof = Ix$. Meaning the composition of the function and its inverse is an identity function. In an inverse function, we may be able to express the independent variable in terms of the dependent variable.

For example, $y = f(x) = \frac{(x + 1)}{x - 1}$
$y = \frac{x + 1}{x - 1}$
$y(x - 1) = x + 1$
$xy - y = x + 1$
$xy - x = 1 + y$
$x(y - 1) = y + 1$
$x = \frac{(y + 1)}{(y - 1)} = g(y)$

Let us find $(gof)(x) = g [ f(x) ]$
$ = g\left [ \frac{(x + 1)}{(x - 1)} \right ]$
= $\frac{\frac{x + 1}{x - 1} + 1}{\frac{x + 1}{x - 1} - 1}$
= $\frac{\frac{x + 1 + x - 1}{x - 1}}{\frac{x + 1 - x + 1}{x - 1}}$=$\frac{2x}{x - 1} \times \frac{x - 1}{2}$
= $x$

Similarly, we can prove that $(fog)(y) = y$, hence $g$ is the inverse of $f$ and $f$ is the inverse of $g$.

Algebraic Functions

A function which consists of finite number of terms involving powers and roots of independent variable x and the four fundamental operations of addition, subtraction, multiplication and division is called an algebraic function.

For example, $f(x) = 5x^{3} - 2x^{2} + 5x + 6$, $g(x) = \frac{\sqrt{(2x + 4)}}{(x - 1)^{2}}$
Polynomials, rational functions and irrational functions are all the examples of algebraic functions.

Transcendental Functions

Functions which are not algebraic are called transcendental functions.

For example, $f(x) = \sin x$, $g(x) = \log (x)$, $h(x) = e^{x}$ , $k(x) = \tan ^{-1} (x)$
Trigonometric functions, Inverse trigonometric functions, exponential functions, logarithmic functions are all transcendental functions.
Types of Functions Graph
The graph along side shows the graph of $y = \sin(x)$

Special Functions

Modulus Functions:

Modulus functions are defined as follows.
y = |x | = { x, if x $\geq$ 0
-x , if x < 0.

For example, | 3 | = 3, and | -4 | = -(-4) = 4, since -4 < 0.

The graph of the modulus function y = |x| is shown below.

Types of Functions Graphs

Greatest Integer Function:

The greatest integer functions are of the form y = [x]. This is also called as the step function.

For example, $[x] = 0$ if $0 \leq x < 1$
$[x] = 1$, if $1 \leq x < 0$

The graph of the greatest integer function will be as shown below. We observe that the function is a curve and is not continuous. Hence, the function is not continuous.

Greatest Integer Function

Constant Function:

If f : R $\rightarrow$ {c}, defined by f(x) = c, for every $x \varepsilon R$, then f is called a constant function of R.

For example, f(x) = 8, g(x) = -5 are all constant functions. The graph of the functions, y = 8 and y = -5 are shown below.

Constant Function

Identity Function:

The identity function maps each member of the domain onto itself. That is, if I is the identity function, then for any x in the domain, I(x) = x. It is also denoted as y = x.

The graph of the identity function is given below.
Identity Function
We observe that the graph passes through the points (-4,-4), (-2,-2), (0,0), (2,2), (4,4) and (6,6). This is because the relation is y = x.

Circular Functions:

The sine and cosine functions of elementary trigonometry and their inverses and functions derived from them.

For example, $f(x) = 2 \sin (x) + 5 \cos (x)$
The graph of the function $y = 2 \sin (x) + 5 \cos (x)$ is given below.

Circular Functions
More topics in Types of Functions
Linear Function Nonlinear Function
Exponential Function Vector Functions
Logarithmic Functions Polynomial Function
Rational Function Square Root Function
Inverse Functions Even and Odd Functions
Piecewise Functions Composite Functions
Sinusoidal Function Identity Function
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