The function \( f(x) = 4(2)^x \) models the number of people sharing a game walkthrough video \( x \) hours after it appears on the gamers page. Let's break it down:
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Initial Value: At \( x = 0 \), the number of people sharing the video is: \[ f(0) = 4(2)^0 = 4(1) = 4 \] This means that 4 people share the video when it first appears.
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Exponential Growth: The function has the form of an exponential function, where the base is 2. This indicates that the number of people sharing the video doubles every hour.
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Rate of Change:
- For each unit increase in \( x \) (each passing hour), the number of people who share the video changes as follows: \[ f(x + 1) = 4(2)^{x + 1} = 4(2^x \cdot 2) = 8(2^x) = 2 \cdot f(x) \]
- Hence, the number of people sharing the video doubles every hour.
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General Behavior: This means that if you know the number of people sharing it at any given hour \( x \), you can predict the number for the next hour by simply multiplying by 2.
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Long-term growth: As \( x \) increases, \( f(x) \) grows rapidly due to the exponential nature of the function, meaning that after some hours, the number of people sharing the video will become very large.
In summary, the function \( f(x) = 4(2)^x \) indicates that the number of people sharing the video doubles every hour, representing exponential growth in the number of sharers over time.