Types of Functions based on Range

Types of Functions based on Range

Inline is the types of function based on the range which is received from the given functions. The different types of functions depending on the range are discussed below. 

Modulus Function

The function f(x)= |x|  is called a modulus function

This can be further defined as

The modulus function returns the absolute value of the given function, irrespective of the sign of the input domain content. The graph of a modulus function extends in the first and the second quadrants as the coordinates of the points on the graph are of the pattern (x, y), (-x,y).                                                                                                                                                                                                                                                          

The domain of |x| is R and its range is [0, ∞).The function f: R →R defined by f(x) = |x| for each x ∈R is called the modulus function. This implies that for every non-negative value of x, f(x) is equivalent to x. Although for negative conditions of x, the value of f(x) is negative concerning the value of x.

Signum Function

Signum function helps determine the sign of the real value function and attributes +1(positive 1) for positive input values of the function, and -1(negative 1) for negative input values of the function.

The signum function simply yields the sign for the assigned values of x. For x value higher than zero, the value assigned to the output is +1, for x value lesser than zero, the value assigned to the output is -1, and for x value equal to zero, the output is equivalent to zero.

The signum function can be interpreted and learned from the below expression.

The signum function f: R→R represented by:

The domain of the signum function covers all the real numbers and is represented along the x-axis, and the range of the signum function has simply two values, +1, -1, drawn on the y-axis.

Domain = R,     Range = {-1, 0, 1} 

Even and Odd Function

The even and odd functions depend on the relationship between the input and the output states of the function. That is for the negative domain value, if the range is also a -ve value of the range of the primary function, then the function is said to be an odd one.

Furthermore for a -ve domain value, if the range is equivalent to that of the primary function, then the function denotes an even one.

In terms of mathematical expression; if f(-x) = f(x), for all the values of x, then the function is considered to be an even function, and if f(-x) = -f(x), for all the given values of x, then the function is said to be an odd function. Consider the below even and odd function examples:

Here, f(-x) = f(x)

Therefore the above function is an even function.

That is; f(x)=-f(-x).

Therefore the function f(x)=x3 is an odd function. 

Inverse Function

Let f: A → B be one-one and onto (bijective) function. Then f-1 exists which is a function f-1 denoted by;   B → A, which maps every component b ∈ B with a component a ∈ A such that f(a) = b is termed as the inverse function of f: A → B.

For inverse of a function the domain and range of the assigned function are reverted concerning the range and domain of the inverse function. The inverse of a function is prominently observed in algebraic functions and inverse trigonometric functions. 

Greatest Integer Function

The function f (x) = [x] is called the greatest integer function and means greatest integer less than or equal to x i.e [x] ≤ x.

OR

The function f: R → R represented by f(x) = [x], x∈R assumes the value of the greatest integer, less or equal to x. Such a function is designated as the greatest integer function.

                            

 The greatest integer function is also recognized as the step function which can be visualized in the above diagram. The domain for such a function is real numbers R, while its range comprises integers (Z).

Periodic Function

The function is thought to be a periodic function if the same range sequentially resembles the different domain values. We can say that the trigonometric functions are periodic. 

For example, the function f(x) = Sinx, has a range equal to the range of [-1, 1] for the various domain values. 

Rational Functions

A Rational function is a sort of function that is derived from the ratio of two given polynomial  Functions and is expressed as,  $f\left(x\right)=\frac{P\left(x\right)}{Q\left(x\right)}$ƒ (x)=P(x)Q(x)      such that P and Q are polynomial functions of x and Q(x) ≠ 0. For example, f(x)=    $f\left(x\right)=\frac{4x+1}{2x+1}$ƒ (x)=4x+12x+1   is a rational function.

Consider the below graph for the rational function given by the equation:

            $f\left(x\right)=\frac{4x+1}{2x+1}$ƒ (x)=4x+12x+1  

                           

The domain and range of rational functions are R. The graphical illustration shows asymptotes, the curves which appear to touch the axes-lines as can be outlined from the above graph.

 

Smallest Integer Functions

The function f (x) = [x] is called the least/smallest integer function and means smallest integer greater than or equal to x i.e [x] ≥ x.

OR

The function f: R → R represented by f(x) = [x], x∈R understands the value of the smallest integer, greater or equal to x. Such a function is designated as the smallest integer function.