The range of the smallest integer function is the set of all integers.
Understanding the Smallest Integer Function
The smallest integer function, also commonly known as the ceiling function, is denoted as [x] or ⌈x⌉. It operates by finding the smallest integer that is greater than or equal to the input real number x.
This definition implies that for any given real number x, the output of [x] will always be an integer that satisfies the condition [x] ≥ x.
Examples of the Smallest Integer Function:
To illustrate how this function works and to better understand its output, consider the following examples:
- For
x = 3.34, the smallest integer greater than or equal to 3.34 is 4. So,[3.34] = 4. - For
x = 0.67, the smallest integer greater than or equal to 0.67 is 1. So,[0.67] = 1. - For
x = 4(an integer itself), the smallest integer greater than or equal to 4 is 4. So,[4] = 4. - For
x = -8.112, the smallest integer greater than or equal to -8.112 is -8. So,[-8.112] = -8. - For
x = -0.5, the smallest integer greater than or equal to -0.5 is 0. So,[-0.5] = 0.
In general, if n is an integer and x is any real number such that n-1 < x ≤ n, then the smallest integer greater than or equal to x will be n.
What is a Function's Range?
In mathematics, the range of a function refers to the set of all possible output values that the function can produce. It contrasts with the domain, which is the set of all possible input values for the function.
Determining the Range of the Smallest Integer Function
As observed from its definition and the examples, the smallest integer function always produces an integer as its output. It never yields a fractional or decimal value.
Furthermore, every integer can be an output of this function. For any integer k, if you input k into the function, [k] = k. If you input any number x slightly less than k (e.g., k - 0.1, k - 0.5, k - 0.99), the function will still output k.
This means that whether the input is positive, negative, or zero, the output will always be an integer. Since the function can produce any integer value, from negative infinity to positive infinity, the range encompasses all integers.
Examples Illustrating the Range
Here's a table summarizing inputs and their corresponding integer outputs:
| Input (x) | Smallest Integer Function [x] (Output) |
|---|---|
| 3.34 | 4 |
| 0.67 | 1 |
| 4 | 4 |
| -8.112 | -8 |
| -0.5 | 0 |
| 5.9 | 6 |
| -2.01 | -2 |
As clearly demonstrated, every output is an integer. Therefore, the range of the smallest integer function is the entire set of integers, which is typically denoted by the symbol ℤ.
