To determine which investment option is better in the long run, we need to analyze the growth of both options over time.

1. Understanding the function $g(x) = 500(1.05)^x$:

   • This represents exponential growth with an initial investment of $500 and a growth rate of 5% per year.

2. Analyzing the first option (graph):

   • Since I can't see the graph, you can check the growth rate visually or use data points from the graph. Compare the slope (growth rate) and the long-term behavior (where the line approaches).

3. Comparative analysis:

   • For the function $g(x)$, as $x$ increases, the value of $g(x)$ will increase exponentially.
   • You may want to evaluate $g(x)$ at several key years (e.g., 0, 5, 10, 20, etc.) to see how the investment grows over time.

4. Long-term growth:

   • Generally, if the graph of the first investment grows slower than the exponential function represented by $g(x)$, then $g(x)$ would be the better investment in the long run.

5. Conclusion:

   • If you can provide more specific details about the graph (like points or the equation representing it), we can analyze it further. Otherwise, based on the exponential growth of $g(x)$, it likely represents a better investment if the first option does not exceed a growth rate of 5% per year.

Would you like to provide more details about the first option or any specific years for comparison? Here are some questions for further exploration:

1. What specific values can you read from the graph for the first investment option?
2. How many years do you want to evaluate both options?
3. Do you know the initial investment amount for the first option?
4. Is there a specific growth rate for the first option indicated in the graph?
5. Would you like a detailed calculation of the growth for different years for both options?

Tip: When comparing investment options, consider both the growth rate and the initial investment to get a clear picture of potential returns.