Types Of Functions

Types Of Functions

1. The functions have a domain X that can be a number, angle, decimal or fraction, and have a range Y of f(x), which is generally a numeric value

2. Based on the set elements

   • Many to One: more than one element has the same co-domain or image f: A → B,
   • Onto Function: every codomain element is related to the domain element and every element in set B has a pre-image in set A f: A → B
   • Into Function: certain elements in the co-domain that do not have any pre-image. The elements in the set B are excess and are not connected to any elements in the set A.
   • Constant Function: all the domain elements have a single image. f (x) = K, where K is a real number
   • One One and Onto Function: every element of the domain is connected to a distinct element in the codomain and every element of the codomain has a pre-image.
   • One one function: every element of the domain has a different image or co-domain element. f: A → B such that every element of set A is connected to a distinct element in set B.

3. Based on equation

   • Identity Function: Has the same domain and range, f(x) = x, or y = x. The graph of the function is a straight line that is equally inclined to the coordinate axes and is passing through the origin and can be positive or negative.
   • Linear Function: The domain and range are real numbers, and it has a straight-line graph, y = x + 2, y = 3x, y = 2x - 1, are all examples of linear functions, y = x can also be considered a linear function. The general form of a linear function is f(x) = ax + b, or y = mx + c.
   • Quadratic function: It has a second-degree quadratic equation, the graph is a non-linear graph and is parabolic in shape, f(x) = ax2 + bx + c, where a ≠ 0 and a, b, c are constants and x is a variable, the domain and range of the quadratic function is R.
   • Cubic function: It has an equation of degree three, f(x) = ax3 + bx2 + cx +d, where a ≠ 0 and a, b, c, and d are real numbers and x is a variable, the domain and range is R.
   • Polynomial function: f(x) = a0xn + a1x n-1 + a2x n-2 + ..... an-1x + an. Here n is a nonnegative integer and x is a variable, the domain and range are R. Based on the power of the polynomial function, the functions can be classified as a quadratic function, cubic function, etc.

4. Based on range

   • Modulus function: Gives the absolute value of the function, is represented as f(x) = |x|. The value of 'x' can be a positive or a negative expression, the graph lies in the first and the second quadrants when the coordinates of the points on the graph are of the form (x, y), (-x, y).
   • Periodic function: It is considered a periodic function if the same range appears for different domain values and in a sequential manner. Also the trigonometric functions can be considered periodicfunctions, for example, the function f(x) = Sinx, have a range equal to the range of [-1, 1] for the different domain values of x = nπ + (-1)nx.
   • Composite function: Is in the form of gof(x), fog(x), h(g(f(x))), and is made from the individual functions of f(x), g(x), h(x). It is made of two functions have the range of one function forming the domain for another function. Let us consider a composite function fog(x), which is made up of two functions f(x) and g(x), we write fog(x) = f(g(x)). The range of g(x) forms the domain for the function f(x). . If f(x) = 2x + 3 and g(x) = x + 1 we have fog(x) = f(g(x)) = f(x + 1) = 2(x + 1) + 3 = 2x + 5.
   • Rational function: Is composed of two functions and expressed in the form of a fraction is a rational function, f(x)/g(x), and g(x) ≠ 0. The functions used in this can be an algebraic function, or any other function.
   • Signum function: It helps us to know the sign of the function but it doesn't give the numeric value or any other values for the range which is limited to {-1, 0, 1}. For the positive value of the domain gives an answer of 1, for negative values gives an answer of -1, and for 0 value of domain, the image is 0.
   • Even and odd functions: They are based on the relationship between the input and the output values of the function. If f(-x) = f(x), for all values of x, then the function is an even function, and if f(-x) = -f(x), for all values of 2 x, then the function is an odd function, some examples of even functions are x , Cosx, Secx, and and some examples of odd functions are x , Sinx, Tanx.
   • Inverse function: The inverse of a function f(x) is f-1(x). For inverse of a function the domain and range of the given function is reverted as the range and domain of the inverse function .An example can be the domain of Sinx is R and its range is [-1, 1], and for Sin-1x the domain is [-1, 1] and the range if R. The inverse of a function exists, if it is a bijective function.
   • Greatest integer function: Is also known as the step function, rounds up the number to the nearest integer less than or equal to the given number. The input variable x can take on any real value, but the output will always be an integer, also, all integers will be in the output set. The domain of this function is real numbers R, while its range will be integers (Z). The greatest integer function graph is known as the step curve because of the step structure of the curve.

5. Based on domain

   • Algebraic function: It has variable, coefficient, constant term and the various arithmetic operators such as addition, subtraction, multiplication, division. f(x) = anxn + an - 1xn - 1+ an-2xn-2+ ax + c. Is also termed as linear function, quadratic function, cubic function, polynomial function, based on the degree of the algebraic equation.
   • Trigononetric function: The six trigonometric functions are f(θ) = sinθ, f(θ) = cosθ, f(θ) = tanθ, f(θ) = secθ, f(θ) = cosecθ. The domain value θ is the angle and is in degrees or in radians, they have been taken based on the ratio of the sides of a right-angle triangle, and are based on the Pythagorastheorem. The domain of the inverse trigonometric function is a real number value and its range is an angle and are also sometimes referred to as periodic functions since the principal values are repeated.
   • Logarithmic functions: They have been derived from the exponential functions and are considered as the inverse of exponential functions. They have a 'log' in the function, and it has a base, y = ?????. The domain value is the input value of 'x' and is calculated using the Napier logarithmic table. It gives the number of exponential times to which the base has raised to obtain the value of x.