To analyze which product's price eventually exceeds all others, let's examine the data provided for both products:
### Product 1:
The price function for Product 1 is given by:
[tex]\[ g(x) = x^2 + 11 \][/tex]
Using the function, we can find the prices after each year:
- Year 1: [tex]\( g(1) = 1^2 + 11 = 12 \)[/tex] dollars
- Year 2: [tex]\( g(2) = 2^2 + 11 = 15 \)[/tex] dollars
- Year 3: [tex]\( g(3) = 3^2 + 11 = 20 \)[/tex] dollars
So, the prices of Product 1 over the three years are [tex]\( [12, 15, 20] \)[/tex] dollars.
### Product 2:
The price function for Product 2 is given by:
[tex]\[ h(x) = 4^x \][/tex]
Using the function, we can find the prices after each year:
- Year 1: [tex]\( h(1) = 4^1 = 4 \)[/tex] dollars
- Year 2: [tex]\( h(2) = 4^2 = 16 \)[/tex] dollars
- Year 3: [tex]\( h(3) = 4^3 = 64 \)[/tex] dollars
So, the prices of Product 2 over the three years are [tex]\( [4, 16, 64] \)[/tex] dollars.
### Comparison:
By examining the prices over the three years, we see that:
- In Year 1, Product 1 has a higher price (12 dollars) compared to Product 2 (4 dollars).
- In Year 2, the prices are close, with Product 1 at 15 dollars and Product 2 at 16 dollars.
- In Year 3, Product 2's price (64 dollars) significantly exceeds Product 1's price (20 dollars).
### Conclusion:
By the third year, the price of Product 2, characterized by an exponential function, clearly exceeds the price of Product 1. Given that exponential growth tends to increase more rapidly than polynomial growth (such as quadratic growth), we can conclude that:
[tex]\[
\text{Product 2 will eventually exceed the price of Product 1 and continue to do so.}
\][/tex]
Thus, the correct answer is:
- Product 2, because the function is exponential.
