Step Function Related Problems

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. A step function f domain is divided or partitioned into a number of intervals. A constant step function f(x) exists in each interval. As a result, the value of the step function does not change within an interval. A step function f, on the other hand, can have different constant values at different intervals. 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.

A common type of step function is the greatest-integer function. The real number set divided into intervals of the form is the domain of the greatest-integer function f. The greatest integer function is a function that rounds up to the nearest integer that is less than or equal to the given number. The greatest integer function has a step curve, which we will investigate in the sections that follow. The greatest integer function has R as its domain and Z as its range. As a result, the greatest integer function simply rounds off to the largest integer that is less than or equal to the given number.

Step function problems

The following are some step function problems with solutions.

Problems: 1 In terms of step functions, write the following function.

if t<6 

if 6 ≤ t < 8 t 

if 8 ≤ t < 30 

if t ≥ 30   

Solution: This function has three unexpected shifts, so we will need three step functions here, one for each shift in the function. In terms of step functions, here is the function.

f(t) = -4 + 29u₆(t) – 9u_s(t) – 6u₃₀(t)

All three step functions are turned off in the first interval, t<6, and the function has the value

f(t) = -4

It’s worth noting that when we know whether step functions are on or off, we usually don’t write them at all, as we did in this case.

The first step function is now on in the next interval, 6 ≤ t < 8 t, while the remaining two are still off. As a result, in this case, the function has a value.

f(t) = -4 + 29 = 25

8 ≤ t < 30  in the third interval The first two step functions are turned on, while the third is turned off. The function has a value here.

f(t) = -4 + 29 – 9 = 16

In the final interval, t ≥ 30  , all three step functions are one, and the function has a value.

f(t) = -4 + 29 – 9 – 6= 10

As a result, the function has the correct value in all intervals.

Problems: 2

Let the function shown be defined for all integers as follows:

y = -2 for x<1 

y = 3 for x≥1 

This function is made up of an infinite number of discrete points, each with a y -coordinate of 2 or 3.

The above graph is viewed as a series of steps, it is also known as a step function graph. In each step, the left endpoint is blocked (light dot) to indicate that the point is a member of the graph, and the other right endpoint is blocked (dark dot) indicates that the values are infinite. That is, only definite values are represented by dark dots.

Step function equation

Step function equation written as:

If y = f(x) = [x] for x ∈ R, a function f: R → R  is called a step or greatest integer function.

Evaluating a step function

Evaluate f(x) = -2(x) for f(-1,5) ,f(9,6) and f(-4)

Solution:
f(-1,5) = -2[(-1.5)]

= -2(-2)

=4

f(9.6) = -2[(9.6)]

= -2(9)

= -18

f(-4) = -2[(-4)]

= 8

Conclusion

In this article, we conclude that, in mathematics, a function on the real numbers is called a step function if it can be written as a finite linear combination of interval indicator functions. 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.