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The function given is $ g(x) = 6(2x + 7) $, which represents the plumbing fee for a local company where $ y $ is the total fee for $ x $ hours worked.

We are asked to find:
1. The total fee after 3 hours.
2. The number of hours for which the total fee will be $282.

### Step 1: Calculate the total fee for 3 hours
Substitute $ x = 3 $ into the function $ g(x) = 6(2x + 7) $:

\[
g(3) = 6(2(3) + 7) = 6(6 + 7) = 6(13) = 78
\]

So, the total fee after 3 hours is **$78**.

### Step 2: Find the number of hours when the total fee is $282
Set $ g(x) = 282 $ and solve for $ x $:

\[
282 = 6(2x + 7)
\]
First, divide both sides by 6:

\[
47 = 2x + 7
\]
Now, subtract 7 from both sides:

\[
40 = 2x
\]
Finally, divide by 2:

\[
x = 20
\]

So, the total fee is $282 after **20 hours**.

### Final Answer:
- The total fee will be **$78** after 3 hours.
- The total fee will be **$282** after 20 hours.

Would you like more details on any step?

Here are 5 related questions to expand your understanding:
1. How does the rate of $6 per hour impact the total fee?
2. What would the total fee be after 10 hours?
3. How do you solve for $ x $ in other similar linear equations?
4. What is the significance of the number 7 in the equation?
5. How can this type of equation be applied in real-world situations?

**Tip**: Always isolate the variable step by step when solving equations like this.

The function g(x) = 6(2x + 7) represents the plumbing fee for a local company where y is the total fee for x hours worked. The total fee will be ___ dollars after 3 hours, and $282 after ___ hours.

Learn how to solve the linear function g(x) = 6(2x + 7) to calculate plumbing fees. This example walks through finding the total fee after 3 hours and determining the hours worked for a total fee of $282. Suitable for Grades 8-10.

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