Liam is using sequences to compare the growth rates of $$ h(x)=1.2 x $$ and $$ j(x)=1.2^{x} $$. Which statement correctly describes how Liam should do this and what he will observe? (1 point) Liam should look at where one sequence has terms greater than the terms in the other sequence. The growth rate of $$ j(x)=1.2^{x} $$ is only greater than the growth rate of $$ h(x)=1.2 x $$ when its terms are greater. Liam should compare the rates of change of the terms in both sequences. The growth rate of $$ h(x)=1.2 x $$ will quickly surpass the growth rate of $$ j(x)=1.2^{x} $$. Liam should compare the rates of change of the terms in both sequences. The growth rate of $$ j(x)=1.2^{x} $$ will quickly surpass the growth rate of $$ h(x)=1.2 x $$. Liam should look at where one sequence has terms greater than the terms in the other sequence. The growth rate of $$ h(x)=1.2 x $$ is greater than the growth rate of $$ j(x)=1.2^{x} $$ when its terms are greater.

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To analyze the growth rates of the functions $$ h(x) = 1.2x $$ and $$ j(x) = 1.2^x $$, we can follow these steps:

1. **Identify the nature of the functions**:
   - $$ h(x) = 1.2x $$ is a linear function. Its growth rate is constant, meaning it increases by a fixed amount (1.2) for each unit increase in $$ x $$.
   - $$ j(x) = 1.2^x $$ is an exponential function. Its growth rate increases as $$ x $$ increases, meaning it grows faster and faster as $$ x $$ becomes larger.

2. **Compare the growth rates**:
   - For small values of $$ x $$, $$ h(x) $$ may be greater than $$ j(x) $$. However, as $$ x $$ increases, $$ j(x) $$ will eventually surpass $$ h(x) $$ due to the nature of exponential growth.
   - To determine where $$ j(x) $$ surpasses $$ h(x) $$, we can set the two functions equal to each other and solve for $$ x $$:
     $$
     1.2x = 1.2^x
     $$

3. **Analyze the rates of change**:
   - The derivative of $$ h(x) $$ is constant: $$ h'(x) = 1.2 $$.
   - The derivative of $$ j(x) $$ is $$ j'(x) = 1.2^x \ln(1.2) $$, which increases as $$ x $$ increases.

4. **Conclusion**:
   - As $$ x $$ increases, the growth rate of $$ j(x) = 1.2^x $$ will surpass the growth rate of $$ h(x) = 1.2x $$. This means that eventually, $$ j(x) $$ will grow much faster than $$ h(x) $$.

Based on this analysis, the correct statement is:

**Liam should compare the rates of change of the terms in both sequences. The growth rate of $$ j(x) = 1.2^{x} $$ will quickly surpass the growth rate of $$ h(x) = 1.2 x $$.**
Liam should compare the rates of change of both sequences. He'll find that the exponential function $$ j(x) = 1.2^{x} $$ grows much faster than the linear function $$ h(x) = 1.2x $$ as $$ x $$ increases.

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