# Greatest Integer Function

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

## Example 1

[*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

[(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.

*-π* = -3.14... is between -3 and -4.

So the greatest integer

that is less than *π* is -4.

So [-*π*] = -4.

## Example 3

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

Lightly draw *y* = *x*^{2} (-2 ≤ *x* ≤ 2).

Quadratic function: vertex form

Draw full circles for each *y* integers on *y* = *x*^{2}.

For *x* = 0, you don't need to draw a circle,

because *y* = *x*^{2} 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* = [*x*^{2}] (-2 ≤ *x* ≤ 2).