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A step function, also known as a staircase function, is a real-number function that can be written as a finite linear combination of interval indicator functions. Informally, a step function is a piecewise constant function with a finite number of pieces. On given intervals, a step function has a constant value, but the constant is different for each interval. The series of horizontal lines is created by the constant value on each interval, and the fact that the constant is different for each interval creates the jumps between each horizontal line segment. This is why a step function’s graph resembles a flight of stairs.
Because this function is a step function, its range is a finite set of values rather than an interval. The function returns four for x values between negative eight and negative two. The function returns negative two for x values between negative two and zero inclusive.
Definition of Step Function
“A step function or staircase function) is a function on the real numbers that can be written as a finite linear combination of interval indicator functions”.
If y = f(x)=[x] for xR, a function f: R →R is called a Step function.
Examples of Step Function
Determine the value of x such that x+1 = 3.
Solution: We have 3 ≤ x+1 < 4 from the definition of the step function .
In this inequality, subtract1.
We get 2 ≤ x < 3 as a result.
Answer: The value of x can be greater than or equal to 2 and less than 3.
Graph of the Step Function
The function of steps Because of the step structure of the curve, the graph is known as the step curve. Consider the expression f(x) = x; if x is an integer, the value of f is x. If x is not an integer, the value of x will be the integer preceding x.
As an example,
- The value of f will be 0 for all numbers in the interval
- f will be set to 1 for the entire interval
- f will take the value 1 for the interval [1,0], and so on
As a result, for an integer n, [n, n+1) will have the step function value as n. The function returns a value that is always between two integers. When the next integer arrives, the function value increases by one unit. This means that the value of f at x = 1 is 1 rather than 0, so there will be a hollow dot at (1,0) and a solid dot at (1,1), where a hollow dot indicates not including the value and a solid dot indicates including the value. These observations lead us to the graph below.
The graph above clearly shows that the function’s inputs can be any real number, but the output will always be integers. As a result, the domain of this function is real numbers (R), and its range is integers ( Z ).
Properties of Step Function
The step function has numerous properties.
- A step function is also the sum or product of two-step functions.
- When a step function is multiplied by a number, the result is another step function. This means that the step functions generates algebra over real numbers.
- A step function has a constant value, but the constant is different for each interval.
- The series of horizontal lines is created by the constant value on each interval, and the fact that the constant is different for each interval creates the jumps between each horizontal line segment.
- A step function can only take a limited number of values.
Conclusion
In this article we conclude that Functions in mathematics describe relationships between two or more quantities. A step function is a type of relationship in which one quantity gradually increases in relation to another quantity. Step functions are used to model idealized physical situations in which some quantity rapidly changes from one value to another in such a way that the exact details of the change are irrelevant for problem solving, such as edges of materials or a process that switches on abruptly at a specific time, etc.
