Definition, domain and range, basic properties
The greatest integer function
Let a be a real number. Then we let \lfloor a\rfloor denote the greatest integer such that \lfloor a\rfloor\leq a. Consider the following function f defined on \mathbb{R} by:
f(x)=\lfloor x\rfloor
It is called the greatest integer function.
Consider the greatest integer function f defined on \mathbb{R} by:
f(x)=\lfloor x\rfloor
Then:
- f(2.1)=\lfloor 2.1\rfloor=2
- f(2)=\lfloor 2\rfloor=2
- f(-2.1)=\lfloor -2.1\rfloor=-3
- f(\pi)=\lfloor \pi\rfloor=3
The domain of the greatest integer function is all real numbers, \mathbb{R}, where the range of the greatest integer function is the set of all integers, \mathbb{Z}.
If a and b are any real numbers then the following hold:
- \lfloor a\rfloor=a if and only if a is an integer;
- If a\leq b then \lfloor a\rfloor\leq \lfloor b\rfloor;
- If n is an integer then \lfloor a+n \rfloor =\lfloor a \rfloor +n
Graphical representation
Graphical representation
The greatest integer function only has integers in its domain. The graph of the greatest integer function is the following:
Equations and inequalities with greatest integer functions
Equations with greater integer functions
If a is an integer then the equation \lfloor x\rfloor=a has the solution set:
a\leq x \lt a+1
If a is a real number which is not an integer then the equation \lfloor x\rfloor=a has no solution.
The equation \lfloor x \rfloor=2 has the solution set [2{,}3).
The equation \lfloor x \rfloor =2.1 has no solutions.
These equations can also be solved graphically.
To solve the equation \lfloor x\rfloor =2 graph the equation y=\lfloor x \rfloor and the horizontal line defined by the equation y=2. The x -values of the points of intersection correspond to the solutions of the equation.
Therefore, the solution set of the equation \lfloor x\rfloor =2 is [2{,}3).
Similarly, the equation \lfloor x \rfloor =2.1 can be seen having no solutions by examining graphs. Graph the function y\longmapsto \lfloor x \rfloor and the horizontal line defined by the equation y=2.1 and observe that there are no points of intersection.
Therefore the equation \lfloor x \rfloor =2.1 has no solutions.
More complex equations can also be solved with the greatest integer function. If f is a function and n an integer then the solution set of the equation \lfloor f\left(x\right)\rfloor =n is:
n\leq f\left(x\right)\lt n+1
Consider the following equation:
\lfloor 3x+1 \rfloor =7
It is solved as follows:
\lfloor 3x+1 \rfloor =7
7\leq 3x+1\lt 8
6\leq 3x\lt 7
2\leq x \lt \dfrac{7}{3}
The solution set is \left[ 2,\dfrac{7}{3} \right).
Inequalities involving the greatest integer function
If a is a real number then the solution set of the inequality \lfloor x\rfloor \leq a is the set of real numbers x such that:
x\lt \lfloor a\rfloor +1
Consider the following inequality:
\lfloor x\rfloor \leq 2.25
The solution set is:
x \lt \lfloor 2.25 \rfloor+1=2+1=3
The inequality \lfloor x \rfloor \leq 2.25 can be solved graphically. Graph the function y\longmapsto \lfloor x \rfloor and shade everything below the horizontal line defined by the equation y=2.25. The x -values of the points on the graph which lie in the shaded area correspond to the solution set of the inequality.
We observe that the solution set is the set of the real numbers x such that:
x \lt \lfloor 2.25 \rfloor+1
Meaning:
x \lt 3
If a is a real number then the solution of the inequality \lfloor x \rfloor \lt a is
- x\lt a if a is an integer
- x\lt \lfloor a \rfloor +1 if a is not an integer
Consider the following inequality:
\lfloor x \rfloor \lt 3
The solution set of the inequality is the set of the real numbers x such that:
x \lt 3
The above rules allow us to solve more complicated inequalities. If f is a function and a is a real number then the solution set of the inequality \lfloor f\left(x\right) \rfloor \lt a is
- f\left(x\right) \lt a if a is an integer
- f\left(x\right) \lt a+1 if a is not an integer
Consider the following inequality:
\lfloor 2x-7\rfloor \lt 3.1
The inequality is solved as follows:
\lfloor-2x-7\rfloor \lt 3.1\\-2x-7 \lt \lfloor 3.1 \rfloor +1\\-2x-7 \lt 3+1=4\\-2x\lt 11\\x\gt \dfrac{-11}{2}
The solution set is \left( \dfrac{-11}{2}, \infty\right).
If a is a real number then the solution set of the inequality \lfloor x \rfloor \geq a is
- x\geq a is a is an integer
- x\geq \lfloor a \rfloor +1 is a is not an integer
Consider the following inequality:
\lfloor x \rfloor \geq 1
The solution set is the set of the real numbers x such that:
x\geq 1
The inequality \lfloor x \rfloor \geq 1 can be solved graphically. Graph the function y\longmapsto \lfloor x \rfloor and shade everything above the horizontal line defined by the equation y=1. The x -values corresponding to the points of the graph in the shaded area correspond to the solution set of the inequality.
We observe that the solution set is the set of the real numbers x such that:
x\geq 1
If a is a real number then the solution set of the inequality \lfloor x \rfloor \gt a is:
x\geq \lfloor a \rfloor +1
Consider the following inequality:
\lfloor x \rfloor \gt 3.1
The inequality is solved as follows:
x\geq \lfloor 3.1 \rfloor +1\\x\geq 4
The solution set of the inequality is \left[ 4,\infty \right).
The above rule allows us to solve more complicated inequalities. If f is a function then the inequality \lfloor f\left(x\right)\rfloor \gt a is equivalent to the inequality
f\left(x\right)\geq \lfloor a\rfloor +1
Consider the following inequality:
\lfloor 2x-1 \rfloor \gt 9.3
The inequality is solved as follows:
\lfloor 2x-1 \rfloor \gt 9.3\\2x-1\geq \lfloor 9.3 \rfloor +1\\2x-1\geq 10\\2x\geq 11\\x\geq \dfrac{11}{2}
The solution set is \left[ \dfrac{11}{2},\infty\right)