Chapter 3.8, Problem 38PPS

To determine

To determine whether the given function $g(x)=\frac{1}{2}〚x〛+4$ is a step function, piecewise function or absolute value function.

Answer to Problem 38PPS

Step function

Explanation of Solution

Given:

Function: $g(x)=\frac{1}{2}〚x〛+4$

Concept used:

A piecewise defined function is a function defined by at least two equations(“pieces”), each of which applies to a different part of the domain.

If the function changes value from one interval to the other, it is a discontinuous function.

Greatest Integer Function is one of the most famous function for step function.

Calculation:

Consider the given function $g(x)=\frac{1}{2}〚x〛+4$ .

Since the given function contains greatest integer function which returns the largest integer less than or equal to $x$ ,

  $g(x)=\frac{1}{2}〚x〛+4$

  $\begin{array}{l}\Rightarrow g(1.7)=\frac{1}{2}〚1.7〛+4=\frac{1}{2}\left(1\right)+4=\frac{9}{2}\\ \Rightarrow g(3.8)=\frac{1}{2}〚3.8〛+4=\frac{1}{2}\left(3\right)+4=\frac{11}{2}\end{array}$

And so on.

Since the value of the function is changing from one interval to the other,

Therefore, this is a step function.

Conclusion:

Therefore, the given function $g(x)=\frac{1}{2}〚x〛+4$ is a step function.