Step Function: Domain, Range, Equation & Examples

A step function is a piecewise constant function with a limited number of pieces in mathematics. It is a linear combination of functions for given intervals. A function on the real numbers, in other words, can be defined as a finite linear combination of indicator functions of given intervals. 

• Step Function is also known as the greatest integer function or the floor function. 
• A discontinuous function is a type of step function. 
• It is used for coordinating session based applications.
• Step Function consists of a series of horizontal line segments arranged in the sequence.
• The value of the constant is different for each value of the function.

Read More: Correlation Coefficient Formula

Key Terms: Step Functions, Functions, Greatest Integer Function, Floor Function, Real Numbers, Domain, Range, Graph

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Step Function

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A step function can be thought of as a finite linear combination of indicator functions for specific intervals. The Greatest Integer Function or Floor Function is another name for a step function. It is a discontinuous function.

• Step functions may be encountered while learning other sorts of functions.
• It includes the sign function "sign(x)," the Heaviside Function, and the Rectangular Function.
• It can, for example, be used to coordinate all of the steps in an e-commerce site's checkout process.
• The following is an example of a step function f: R R:

\(f (x) = \sum_{i = 0}^n a_iX_{Ai} (X)\)

• The real numbers are defined by x in the above equation.
• A is defined for the interval with the constraint n >= 0 and is a real number.
• If this requirement is met, then XA will provide the indicator function A.
• X_A Is given by

• If x belongs to A, the value of the function is 1.
• In case x does not belong to A, the value is 0.
• When x is a real number and y = f(x) = xx, a function provided by f: RR is referred to as the greatest integer function.

Read More: Increasing and Decreasing Functions in Calculus

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What is Unit Step Function?

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A Heaviside function is also known as a unit step function. After the provided time interval specified by t, the value of the given function continues to change. The unit step function is commonly indicated by u (t), which is further denoted by the unit step signal below:

• The function of time is defined as u in the formula above. 
• The value of u(t) is 0 when time turns negative. 
• If time is positive, however, the value is 1.
• In the case of the above equation, the graph will be as follows:
• The equation mentioned above is satisfied by the graph of the unit step function. 

u = f (t) (t)

Read More: Linear Regression Formula

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Derivation of Step Function 

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Step Function works for all levels of function except for the situation of t =0. As a result, the derivative of the step function is zero for all t values. When t = 0, however, it becomes limitless.

• The impulse function is the derivative of the unit step function.
• Impulse functions are used by engineers to create models for certain events.
• In most circumstances, however, the value of the impulse function is zero.

Read More: Types of Functions

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Domain and Range

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Domain of any function refers to the range of input values that can be used for an independent variable in the function. On the other hand, the range of a function is the set of output values generated for the function's domain (input values). In the case of a step function, f(x) takes the value of the biggest integer less than or equal to x for each value of x. Consider the following scenario:

• -3 x [-2.19] = -2.19
• 3 = [3.67]
• -1 = [-0.83]

This function's domain is a set of real integers separated into intervals like [-5, 3], [-4, 2), [-3, 1), [-2, 0), and so on. This shows how a step function's domain and range are related. This can be summarised as follows:

• -2 x -1 [x] = -2, -2 x -1
• -1 x 0 [x] = -1, -1 x 0
• [x] = 0 x 1 x 1 x 1 x 1 x 1 x 1 x 1 x 1 x 1 x
• 1 x 2 = 1 x 2 = 1 x 2 = 1 x 2 = 1 x 2 = 1 x 2 = 1 x

Read More: Domain, Co Domain, Range

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Step Function Graph

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Drawing a graph for a step function is comparable to drawing a graph for any piecewise function. Each portion of this function must be graphed separately. The steps for graphing a step function are as follows:

• To begin, draw a horizontal line segment at each constant output value.
• Each of these value corresponds to the input values.
• The second step is to create a closed circle point on each horizontal line at the included endpoint.
• This indicates that if the end value is included in that interval, it should be represented by a filled circle.
• Finally, at each horizontal line's endpoint, draw an open circle point (a circle that hasn't been filled in).
• To put it another way, if the final value is not included in that interval, it should be represented as an open circle.

Read More: Derivatives 

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Things to Remember

• A step function is the sum or product of two-step functions. 
• When a step function is multiplied by a number, the outcome is a step function again.
• As a result, the step functions produce algebra over real numbers.
• Only a limited number of values can be taken by a step function. 
• The definite integral of a step function is a piecewise linear function.

Read More: Mean Absolute Deviation