Q1. - Algebra 2

Question

Find a counterexample to the statement to find the greatest integer function of x when x is not an integer, round x to the nearest integer.

Step-by-Step Solution

Answer

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

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 $\left[\right[2.6\left]\right]=2$.