Let A = {7, 8, 9, 10, 11, 12, 13; 14, 15, 16} and let f ∶ A → N be defined by f(x) = the highest prime factor of x. How many elements are there in the range of f?
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Finding the Range of a Function Based on Highest Prime Factor
The problem asks us to determine the number of elements in the range of a function f, defined over a specific set A. The function f(x) gives the highest prime factor of the element x.
First, let's list the elements of the set A:
\(A = \{7, 8, 9, 10, 11, 12, 13, 14, 15, 16\}\)
The function is defined as \(f: A \rightarrow N\), where \(f(x) = \text{the highest prime factor of } x\).
The range of the function f is the set of all possible values of f(x) for each x in the domain A. To find the range, we need to calculate f(x) for every element in A and then collect the unique values.
Let's calculate the highest prime factor for each element in A:
| Element \(x\) | Prime Factorization of \(x\) | Highest Prime Factor \(f(x)\) |
|---|---|---|
| 7 | 7 | 7 |
| 8 | \(2 \times 2 \times 2\) | 2 |
| 9 | \(3 \times 3\) | 3 |
| 10 | \(2 \times 5\) | 5 |
| 11 | 11 | 11 |
| 12 | \(2 \times 2 \times 3\) | 3 |
| 13 | 13 | 13 |
| 14 | \(2 \times 7\) | 7 |
| 15 | \(3 \times 5\) | 5 |
| 16 | \(2 \times 2 \times 2 \times 2\) | 2 |
The values of f(x) for \(x \in A\) are {7, 2, 3, 5, 11, 3, 13, 7, 5, 2}.
The range of f is the set of these values, considering only the unique elements. Let's list the unique values:
- 2
- 3
- 5
- 7
- 11
- 13
The set of unique values in the range of f is {2, 3, 5, 7, 11, 13}.
To find the number of elements in the range, we simply count the number of unique values we found.
Number of elements in the range = 6.
Therefore, there are 6 elements in the range of the function f.
Revision Table: Function Range and Prime Factors
| Concept | Description | Example |
|---|---|---|
| Domain | The set of all possible input values for a function. | For \(f(x)\) in this problem, the domain is set A. |
| Range | The set of all possible output values of a function. | For \(f(x)\) in this problem, the range is the set of unique highest prime factors of elements in A. |
| Prime Factor | A prime number that divides a given number exactly. | Prime factors of 12 are 2 and 3. |
| Highest Prime Factor | The largest prime number that divides a given number. | Highest prime factor of 12 is 3. |
Additional Information: Understanding Functions and Number Theory
This problem combines concepts from set theory (understanding sets and functions) and number theory (prime factorization).
- A function maps each element in its domain to exactly one element in its codomain. The set of all actual output values is called the range.
- Prime numbers are fundamental building blocks in number theory. A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself (e.g., 2, 3, 5, 7, 11, 13, ...).
- Every integer greater than 1 can be uniquely represented as a product of prime numbers (Fundamental Theorem of Arithmetic). Finding the highest prime factor involves performing this factorization.
- When determining the range of a function, it is crucial to list all output values and then identify only the unique ones, as a set only contains distinct elements.
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