The SQL operator used to check if a value falls within a specified range is the BETWEEN operator.

The BETWEEN operator is a logical operator in SQL that is used to check if a value falls within a specified range. It is commonly used in SQL queries to filter data based on a range of values.

Syntax:
The syntax for using the BETWEEN operator in SQL is as follows:

```
SELECT column_name(s)
FROM table_name
WHERE column_name BETWEEN value1 AND value2;
```

Explanation:
- The BETWEEN operator is used in the WHERE clause of an SQL query to specify a range of values for a particular column.
- The column_name represents the column in the table that you want to check for values within the specified range.
- The value1 and value2 represent the lower and upper bounds of the range, respectively.
- The BETWEEN operator checks if the column value is greater than or equal to value1 and less than or equal to value2.
- If the column value satisfies the condition, it is included in the result set; otherwise, it is excluded.

Example:
Let's consider a table called "employees" with columns "name" and "salary". We want to retrieve all employees whose salary falls between $2000 and $5000.

```
SELECT name, salary
FROM employees
WHERE salary BETWEEN 2000 AND 5000;
```

This query will return all the employees whose salary is greater than or equal to $2000 and less than or equal to $5000.

Conclusion:
In conclusion, the correct SQL operator used to check if a value falls within a specified range is the BETWEEN operator. It is a powerful operator that allows us to filter data based on a range of values. By using the BETWEEN operator in the WHERE clause of an SQL query, we can easily retrieve the desired results.