Graph Of A Step Function

In Mathematics, a pace function (also called equally staircase function) is divers as a piecewise constant office, that has simply a finite number of pieces. In other words, a role on the real numbers can be described as a finite linear combination of indicator functions of given intervals. It is also chosen a floor function or greatest integer function. The footstep part is a discontinuous part. Even so, a mathematical definition of a step function is given beneath forth with an example.

Step Function Definition

A step function f: R → R can be written in the course:

Step function formula 1

for all existent numbers x.

If northward ≥ 0, α_i are existent numbers and A_i are intervals, then the indicator function of A is χ_A, and it tin can be written equally beneath:

Step function formula 2

(Or)

A function f: R → R is called a footstep or greatest integer role if y = f(x) = [ten] for x ∈ R.

Domain and Range

Every bit nosotros know, for whatsoever function domain is referred to every bit the ready of input values that tin can be taken for an contained variable in the given function. Range of a function is divers every bit the set of output values generated for the domain (input values) of the function. In the example of a step function, for each value of x, f(x) takes the value of the greatest integer, less than or equal to x. For instance:

[-2.xix] = -3 [iii.67] = 3 [-0.83] = -i

The domain of this office is a grouping of real numbers that are divided into intervals such as [-five, three), [-4, 2), [-3, 1), [-2, 0) and and then on. This explains the domain and range relations of a step office.

This tin can be generalized every bit given below:

[x] = -2, -2 ≤ x < -1 [x] = -1, -one ≤ x < 0 [x] = 0, 0 ≤ ten < i [x] = i, 1 ≤ x < two

Footstep Function Graph

The process of drawing a graph for a stride part is similar to the graphing any piecewise role. Here, we need to graph each part of this function individually. The steps required to graph a step role are given below:

Offset, depict a horizontal line segment at each constant output value throughout input values that information technology corresponds.
The second step is to draw a closed circle bespeak (a filled-in circle) at the included endpoint on each horizontal line that ways if the end value is included in that interval, then it should be represented with a filled circle.
Finally, draw an open circle bespeak (a circumvolve non filled in) at the endpoint that is excluded on each horizontal line. In other words, if the cease value is non included in that interval, so information technology should exist represented with an open circle.

Thus, the graph of a step part is every bit shown beneath.

Step function graph

Pace Office Case

Question

Draw a graph of the pace function:

Example function 1

Solution:

From the given,

-two, 0, ii are the values of y.

10 < -1 ways the values of 10 = …, -iv, -3, -two, -1

-1 ≤ ten ≤ two means, ten = -1, 0, 1, two

x > i ways the values of ten = 1, 2, 3, iv, …..

Past plotting these values on a graph newspaper, the below graph volition obtain.

Step function graph 2

The above graph is viewed as a group of steps and hence it is also called a stride function graph. The left endpoint in every stride is blocked(dark dot) to show that the point is a fellow member of the graph, and the other right end arrow indicates that the values are infinite. That ways but definite values are shown with dark dots.

Backdrop

The important properties of pace functions are given beneath:

The sum or production of ii-stride functions is likewise a step function.
If a step function is multiplied by a number, then the event produced is once again a pace function. That indicates the stride functions create an algebra over the existent numbers
A step function can take simply a finite number of values
Piecewise linear function is the definite integral of a step role

One tin discover pace function when dealing with other types of functions for different reasons. For instance, a lilliputian case of a step function is a constant function. The simplest non-abiding step function is sign function "sgn(x)", considering this function results -1 for the negative inputs and 1 (i.east. +1) for the positive input values. Other types include Heaviside function and rectangular function, etc.

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