It looks like there might be a small mistake in the equation you've provided. The equation \( f(x) = 40x + 57 \) seems to be a more appropriate representation of the function. Here's how to interpret it:
- \( f(x) \) represents the total number of stamps in Alice's collection.
- \( x \) is the number of weeks that have passed since September 1.
- The term \( 40x \) suggests that Alice is acquiring 40 new stamps each week.
- The \( +57 \) indicates that Alice starts with 57 stamps in her collection.
Function Interpretation
The function can be interpreted as:
- Initial Stamps: 57 (the starting point).
- Rate of Increase: 40 stamps per week.
Finding the Range
To determine the range of this function, we need to consider the values of \( x \):
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Domain: Since \( x \) represents the number of weeks since September 1, \( x \) will be \( 0 \) (on September 1) and will continue increasing as time goes on.
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The minimum value occurs at \( x = 0 \): \[ f(0) = 40(0) + 57 = 57 \]
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As \( x \) increases, \( f(x) \) will continue to increase: \[ f(x) = 40x + 57 \quad \text{will approach infinity as } x \text{ increases.} \]
Conclusion
Thus, the range of the function \( f(x) = 40x + 57 \) is:
\[ [57, \infty) \]
This means that the total number of stamps in Alice's collection will start at 57 and can increase indefinitely as he continues to buy stamps each week.