Let's examine the given information carefully to determine which product's price eventually exceeds all others and why.
The table provided shows the prices of two products across three years:
[tex]\[
\begin{tabular}{|l|l|l|l|l|}
\hline \text{Product} & \text{Function} & \begin{tabular}{c} \text{Year 1} \\ \text{(dollars)} \end{tabular} & \begin{tabular}{c} \text{Year 2} \\ \text{(dollars)} \end{tabular} & \begin{tabular}{c} \text{Year 3} \\ \text{(dollars)} \end{tabular} \\
\hline \text{Product 1} & \( g(x) = x^2 + 11 \) & 12 & 15 & 20 \\
\hline \text{Product 2} & \( h(x) = 4x \) & 4 & 16 & 64 \\
\hline
\end{tabular}
\][/tex]
Step-by-Step Analysis:
1. Identify Functions and Yearly Values:
- Product 1: [tex]\( g(x) = x^2 + 11 \)[/tex]
- Product 2: [tex]\( h(x) = 4x \)[/tex]
- Yearly values for Product 1 are:
- Year 1: 12 dollars
- Year 2: 15 dollars
- Year 3: 20 dollars
- Yearly values for Product 2 are:
- Year 1: 4 dollars
- Year 2: 16 dollars
- Year 3: 64 dollars
2. Compare Yearly Values Directly:
- Year 1:
- Product 1: 12 dollars
- Product 2: 4 dollars
- Product 1 has the higher value.
- Year 2:
- Product 1: 15 dollars
- Product 2: 16 dollars
- Product 2 has the higher value.
- Year 3:
- Product 1: 20 dollars
- Product 2: 64 dollars
- Product 2 has the higher value.
3. Analyze the Growth Functions:
- The function [tex]\( g(x) = x^2 + 11 \)[/tex] for Product 1 is quadratic, meaning it grows at a polynomial rate.
- The function [tex]\( h(x) = 4x \)[/tex] for Product 2 is linear, meaning it grows at a constant rate.
4. Determine which product's price eventually exceeds all others:
Based on the values from the table:
- In Year 3, Product 2 has a value of 64 dollars, which is significantly higher than Product 1's value of 20 dollars.
Therefore, the analysis shows that:
Product 2 eventually exceeds Product 1 in price due to its higher value in the later years.
Conclusion:
Product 2, because it has a higher Year 3 value.
