1. Let's state the problem: We want to differentiate between the expressions "Multiply n by two, then add three" and "Add three to n, then multiply by two". 2. First expression: "Multiply n by two, then add three" can be written as the function: $$f(n) = 2n + 3$$ 3. Second expression: "Add three to n, then multiply by two" can be written as: $$g(n) = 2(n + 3)$$ 4. Let's simplify the second function: $$g(n) = 2n + 6$$ 5. Notice the difference between the two functions: - First function: $$f(n) = 2n + 3$$ - Second function: $$g(n) = 2n + 6$$ 6. The difference lies in the constant term added after multiplication. 7. To understand the difference in their graphs, note that both are linear functions with the same slope 2, but different y-intercepts (3 and 6 respectively). 8. This means the graph of $$g(n)$$ is shifted upward by 3 units compared to $$f(n)$$. 9. In summary, the order of operations affects the constant term in the function, which shifts the graph vertically. Final answer: The two expressions correspond to $$f(n) = 2n + 3$$ and $$g(n) = 2n + 6$$, differing by a vertical shift of 3 units.