Greatest Integer Function

Greatest Integer Function

How to solve the greatest integer function problems ([x]): examples and their solutions.

Example 1

Find the value of each expression. 1. [2], 2. [2.7], 3. [pi]

[a] means
'the greatest integer
that is less than or equal to a'.

[(integer)] means
the greatest integer
that is less than or equal to (integer).

So [2] = 2.

2.7 is between 2 and 3.

So the greatest integer
that is less than 2.7 is 2.

So [2.7] = 2.

π = 3.14... is between 3 and 4.

So the greatest integer
that is less than π is 3.

So [π] = 3.

Example 2

Find the value of each expression. 1. [-2], 2. [-0.6], 3. [-pi]

[(integer)] means
the greatest integer
that is less than or equal to (integer).

So [-2] = -2.

-0.6 is between 0 and -1.

So the greatest integer
that is less than -0.6 is -1.

So [-0.6] = -1.

= -3.14... is between -3 and -4.

So the greatest integer
that is less than π is -4.

So [-π] = -4.

Example 3

Graph the given function. y = [x].

Lightly draw y = x.

Draw full circles for each y integers on y = x.

Draw empty circles beneath each full circle.
(1 unit below)

Draw horizontal segments
whose endpoints are the full circle and the empty circle.

The blue graph shows y = [x].

Just like this graph,
a function whose y values change like a step
is called a 'step function'.

y = [x] is an example of a step function.

Example 4

Graph the given function. y = [x^2] (-2 <= x <= 2).

Lightly draw y = x2 (-2 ≤ x ≤ 2).

Quadratic function: vertex form

Draw full circles for each y integers on y = x2.

For x = 0, you don't need to draw a circle,
because y = x2 doesn't go under y = 0 near x = 0.

Draw empty circles beneath each full circle.
(1 unit below)

Draw horizontal segments
whose endpoints are the full circle and the empty circle.

The blue graph shows y = [x2] (-2 ≤ x ≤ 2).