To determine the image (or range) of the function [tex]\(\beta\)[/tex] defined by [tex]\(\beta: \mathbb{Z}^+ \rightarrow \mathbb{Z}^+\)[/tex] with [tex]\(\beta(x) = 2x + 1\)[/tex], we should analyze the outputs of the function for inputs belonging to the positive integers [tex]\(\mathbb{Z}^+\)[/tex].
### Step-by-Step Solution:
1. Understand the function definition:
- The function [tex]\(\beta\)[/tex] maps positive integers to positive integers.
- For any positive integer [tex]\(x\)[/tex], the function outputs [tex]\(2x + 1\)[/tex].
2. Calculate the function's output for the first few positive integers:
- When [tex]\(x = 1\)[/tex]:
[tex]\[
\beta(1) = 2(1) + 1 = 3
\][/tex]
- When [tex]\(x = 2\)[/tex]:
[tex]\[
\beta(2) = 2(2) + 1 = 5
\][/tex]
- When [tex]\(x = 3\)[/tex]:
[tex]\[
\beta(3) = 2(3) + 1 = 7
\][/tex]
- When [tex]\(x = 4\)[/tex]:
[tex]\[
\beta(4) = 2(4) + 1 = 9
\][/tex]
- When [tex]\(x = 5\)[/tex]:
[tex]\[
\beta(5) = 2(5) + 1 = 11
\][/tex]
3. Infer the pattern:
- The outputs obtained from the calculations are 3, 5, 7, 9, 11.
- These numbers suggest a pattern of being odd numbers, starting from 3 and increasing by 2 for each successive positive integer input.
4. Generalize the image (range) of the function:
- For any positive integer [tex]\(x\)[/tex], [tex]\(\beta(x) = 2x + 1\)[/tex] always yields an odd number because multiplying any integer by 2 results in an even number, and adding 1 to an even number results in an odd number.
- Therefore, the image of [tex]\(\beta\)[/tex] is the set of all positive odd numbers greater than or equal to 3.
### Conclusion:
The image (or range) of the function [tex]\(\beta\)[/tex] defined by [tex]\(\beta(x) = 2x + 1\)[/tex] for [tex]\(x \in \mathbb{Z}^+\)[/tex] is the set of all positive odd numbers starting from 3. Explicitly, the first few elements in this set are 3, 5, 7, 9, 11, and so on.
[tex]\[
\boxed{\{3, 5, 7, 9, 11, \ldots\}}
\][/tex]
