Question

How do you evaluate the function with the given value of x :
$f\left( x \right)=\dfrac{1}{2}\left[ x+1 \right]$ ; x = 0; x = -1

Answer 1

$\left[ x \right]$ is the greatest integer function of x , the value of $\left[ x \right]$ is the highest integer which is less than equal to x. so [ 3 .6] is equal to 6 and [ -2.4 ] is equal to – 3. The value of $\left[ x \right]$ is equal to x when x is an integer for example [3] is equal to 3.

Complete step-by-step answer:
We have evaluate $f\left( x \right)=\dfrac{1}{2}\left[ x+1 \right]$ at x = 0 and at x = -1
If we put x equal to 0 in the equation $f\left( x \right)=\dfrac{1}{2}\left[ x+1 \right]$ we will get $\dfrac{1}{2}\left[ 1 \right]$ the value of [1] is equal to 1 because $\left[ x \right]$ is equal to x when x is an integer. So $\dfrac{1}{2}\left[ 1 \right]$ will be equal to $\dfrac{1}{2}$
If we put – 1 in the function $f\left( x \right)=\dfrac{1}{2}\left[ x+1 \right]$ we will get $\dfrac{1}{2}\left[ 0 \right]$.
The value of $\dfrac{1}{2}\left[ 0 \right]$ is equal to 0 because we know that [ 0] is equal to 0.
So at x equal to 0 the function value is $\dfrac{1}{2}$ and at x equal to – 1 the function value is 0.

Note: The greatest integer function is not a continuous function , it is also called a step function because the graph of the greatest integer function is like a step. The limit value of the function does not exist at every integer. The slope of the function is 0 at every value of x except at the integers.