a certain brand of cereal is packaged so that the mass of the cereal in one box is 504 grams. Five servings of the cereal is equivalent to 140 grams of cereal.  Explain how you could solve f^-1(w)=10 for w using function f instead of f^-1

To solve \( f^{-1}(w) = 10 \) for \( w \) using the function \( f \), let's first understand what the inverse function \( f^{-1} \) represents and how it relates to the original function \( f \).

### Understanding the Problem

Given:
- A box of cereal weighs 504 grams.
- Five servings of cereal weigh 140 grams.

From this information, we can define a function \( f \) that converts the number of servings into the corresponding mass in grams.

#### Step 1: Define the Function \( f \)

We know that:
- \( 5 \) servings of cereal correspond to \( 140 \) grams.

The weight of one serving \( w_s \) is calculated as:
\[
w_s = \frac{140 \text{ grams}}{5 \text{ servings}} = 28 \text{ grams per serving}
\]

So, the function \( f \) can be defined as:
\[
f(x) = 28x
\]
where \( x \) is the number of servings, and \( f(x) \) gives the mass in grams of \( x \) servings.

#### Step 2: Understanding the Inverse Function \( f^{-1} \)

The inverse function \( f^{-1}(w) \) represents the number of servings for a given weight \( w \). 

If we need to solve \( f^{-1}(w) = 10 \), this tells us that there are 10 servings corresponding to some weight \( w \).

### Solving \( f^{-1}(w) = 10 \) for \( w \) Using \( f \)

To solve \( f^{-1}(w) = 10 \), we are essentially looking for the value of \( w \) such that \( 10 \) servings weigh \( w \) grams.

Using the definition of \( f \), we know:
\[
f(10) = 28 \times 10 = 280
\]

This means that 10 servings weigh \( 280 \) grams.

Therefore, the value of \( w \) when \( f^{-1}(w) = 10 \) is:
\[
w = 280
\]

### Conclusion

To find \( w \) using the original function \( f \), we:
1. Recognized that \( f^{-1}(w) = 10 \) implies \( 10 \) servings of cereal correspond to \( w \) grams.
2. Used the function \( f(x) = 28x \) to calculate the weight of \( 10 \) servings.
3. Computed \( f(10) = 280 \), concluding that \( w = 280 \).

Thus, \( w = 280 \) grams when \( f^{-1}(w) = 10 \).

### Would you like further details or have any questions?

Here are some related questions for further exploration:

1. How would you define the function \( f^{-1} \) explicitly if given \( f(x) = 28x \)?
2. What is the general formula for the inverse of a linear function?
3. If the weight per serving changed, how would the function \( f(x) \) and its inverse \( f^{-1}(x) \) change?
4. How would you solve \( f^{-1}(w) = 15 \) using the function \( f \)?
5. Can you calculate the number of servings if given a different weight, say 560 grams, using the function \( f^{-1} \)?

**Tip:** Understanding the relationship between a function and its inverse is crucial for solving equations involving inverses.

Explore how to solve the equation f^-1(w) = 10 using the function f(x) = 28x. Understand the relationship between servings of cereal and their corresponding weight in grams. Suitable for high school students and anyone interested in functions and inverse functions.

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