The table provides values of the function $f(x) = 7^x$ for $x$ values approaching $\pi$. We need to approximate $7^\pi$ to four decimal places, using the closest approximation of $\pi \approx 3.14159265359$.

Let's look at the table:

• The value closest to $\pi$ is $x = 3.14159265$.
• The corresponding value of $f(x) = 7^x$ is $451.8079$.

Thus, the approximation of $7^\pi$, correct to four decimal places, is:

$7^\pi \approx 451.8079$

Would you like more details on this problem, or do you have any questions?

Here are 5 related questions to expand on the topic:

1. How is the value of $\pi$ determined accurately in mathematics?
2. Why does $7^x$ grow so quickly compared to linear functions?
3. How can we calculate powers of irrational numbers like $\pi$ more precisely?
4. What methods are available for approximating the value of $f(x)$ when exact computation isn't possible?
5. What are some real-world applications where calculating powers involving $\pi$ is important?

Tip: When approximating functions using tables, try to find values that are very close to the target to minimize error.