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The population f(x), in millions, of State A of a country after x years is represented by the function shown below:

f(x) = 4(1.08)x

The graph shows the population g(x), in millions, of State B of the country after x years:

graph of exponential function g of x that curves up from left to right and goes through points 0 comma 2 and 9 comma 4

Which conclusion is correct about the populations of State A and State B? (1 point)

The original population of State A was double the original population of State B.

The original population of State B was double the original population of State A.

The original population of State A was four times the original population of State B.

The original population of State A was equal to the original population of State B.


To solve this problem, we need to analyze the given function for State A and the graph for State B to compare their original populations.

### Step 1: Analyze State A
The population of State A is given by the function:
\[ f(x) = 4 \times (1.08)^x \]
The original population of State A (at \( x = 0 \)) can be calculated by substituting \( x = 0 \) into the function:
\[ f(0) = 4 \times (1.08)^0 = 4 \times 1 = 4 \]
So, the original population of State A is 4 million.

### Step 2: Analyze State B
From the graph, we can observe the population of State B. The graph passes through the point \( (0, 2) \), meaning that the population at \( x = 0 \) is 2 million.

### Step 3: Compare the Original Populations
- The original population of State A is 4 million.
- The original population of State B is 2 million.

Since 4 million is double 2 million, the correct conclusion is:

**The original population of State A was double the original population of State B.**

### Do you want any further details or have any questions?

### Related Questions:
1. How do exponential functions compare in terms of growth rate?
2. What is the significance of the base in an exponential function?
3. How does the graph of an exponential function change as the base increases?
4. What other real-life scenarios can be modeled using exponential functions?
5. How can we calculate the doubling time of a population using an exponential function?

**Tip:** When analyzing exponential growth, always check the initial value (when \( x = 0 \)) and the base of the exponent to understand the rate of growth.

Explore the comparison of State A and State B populations using exponential growth analysis. Understand how to calculate original populations and analyze growth rates with detailed step-by-step solutions. Suitable for grades 9-12.

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