Question

determine the range of the function model by the equation in late August Alice is given some stamps by his uncle on September 1st he begins to buy new stamps to add to his collection. The number of stamps in his collection can be determaned using the equation f(x)=40x=57, in which x indicates the number of weeks that have gone by since September 1 and f(x) represents the total number of his stamps.+

Answer 1

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 \( x \):

1. **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.

2. The minimum value occurs at \( x = 0 \):
\[
f(0) = 40(0) + 57 = 57
\]

3. 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.