Question

Find a counterexample to the statement to find the greatest integer function of xwhen xis not an integer, round xto the nearest integer.

Short answer

The counter example for the given statement is $\left[\right[2.6\left]\right]=2$

Step-by-step solution

Step-1 – Greatest integer function

The Greatest integer function, written $f\left(x\right)=\left[\left[x\right]\right]$is an example of step function. The symbol$\left[\left[x\right]\right]$ means the greatest integer less than or equal to x

Step-2 – Suitable function for given statement.

The given statement is to find the greatest integer function of x when x is not an integer, round x to the nearest integer.

Suitable function for this case is $g\left(x\right)=\left[\left[x\right]\right]$

$\begin{array}{l}Letusconsiderx=2.6\\ g\left(x\right)=〚x〛\\ g\left(2.6\right)=〚2.6〛\\ =2\end{array}$

Step-3 – Determine the difference.

Original function was:

$\begin{array}{l}x=2.6\\ 2<2.6<3\end{array}$

Here, we can observe that to find the greatest integer function of x when x is not an integer, we didn’t round x to the nearest integer. We round to integer less than x.

So, the counter example for the given statement is $\left[\right[2.6\left]\right]=2$.