To determine which product's price will eventually exceed all others, we need to analyze the nature of the functions that represent each product's price over time, as well as the given data.
### Analysis of Product Functions
- Product 1:
- Function: [tex]\( g(x) = 2^x \)[/tex]
- This is an exponential function which typically grows rapidly as [tex]\( x \)[/tex] increases.
- Product 2:
- Function: [tex]\( h(x) = x^2 + 12 \)[/tex]
- This is a quadratic polynomial function, which grows relatively slower than an exponential function as [tex]\( x \)[/tex] increases.
### Yearly Prices
Let's consider the prices given for each product over the first three years:
| Year | Product 1 Price | Product 2 Price |
|------|---------------------|------------------|
| 1 | [tex]\( g(1) = 2 \)[/tex] | [tex]\( h(1) = 13 \)[/tex] |
| 2 | [tex]\( g(2) = 4 \)[/tex] | [tex]\( h(2) = 16 \)[/tex] |
| 3 | [tex]\( g(3) = 8 \)[/tex] | [tex]\( h(3) = 21 \)[/tex] |
From the table, we see:
- Year 1:
- Product 1: 2 dollars
- Product 2: 13 dollars
- Year 2:
- Product 1: 4 dollars
- Product 2: 16 dollars
- Year 3:
- Product 1: 8 dollars
- Product 2: 21 dollars
### Comparing Growth Rates
Even though Product 2 starts with much higher prices than Product 1, we need to pay attention to how each function grows:
- The exponential function [tex]\( 2^x \)[/tex] of Product 1 will eventually outpace the polynomial [tex]\( x^2 + 12 \)[/tex] function of Product 2, because, in general, exponential growth increases faster than polynomial growth as [tex]\( x \)[/tex] becomes larger.
### Conclusion
Given the rapid growth of exponential functions compared to polynomial functions, it is clear that the price of Product 1, represented by the function [tex]\( g(x) = 2^x \)[/tex], will eventually exceed the price of Product 2, represented by the function [tex]\( h(x) = x^2 + 12 \)[/tex], for sufficiently large values of [tex]\( x \)[/tex]. Therefore, the correct conclusion is:
- Product 1, because the function is exponential.
