A gas tank can hold 12 gallons of gas. The equation $$ p(g)=3.5 g $$ gives the amount of money it costs when filling the tank with gas. What is the range for this situation? $$ \{0,1,2,3, \ldots, 42\} $$ $$ \{3.50,7,10.50,14, \ldots 42\} $$ $$ \{p(g) \mid 0 \leq p(g) \leq 42\} $$ $$ \{p(g) \mid p(g) \leq 42\} $$

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To determine the **range** of the function $$ p(g) = 3.5g $$ given that the gas tank can hold up to **12 gallons**, let's analyze the situation step-by-step.

### Understanding the Function

- **Function:** $$ p(g) = 3.5g $$
  - Here, $$ p(g) $$ represents the cost in dollars.
  - $$ g $$ represents the number of gallons of gas.

- **Domain:** Since the gas tank can hold a maximum of 12 gallons, $$ g $$ can range from **0 to 12**.
  - $$ 0 \leq g \leq 12 $$

### Calculating the Range

- **Minimum Cost:** When $$ g = 0 $$ gallons,
  $$
  p(0) = 3.5 \times 0 = \$0
  $$

- **Maximum Cost:** When $$ g = 12 $$ gallons,
  $$
  p(12) = 3.5 \times 12 = \$42
  $$

- **Intermediate Costs:** For any value of $$ g $$ between 0 and 12, $$ p(g) $$ will yield a cost between \$0 and \$42. Since $$ g $$ can take any real number within this interval, the costs are **continuous** and not just discrete values.

### Evaluating the Options

1. **$$ \{0,1,2,3, \ldots, 42\} $$:**
   - **Incorrect.** This suggests only integer dollar amounts, ignoring the continuous nature of gas measurements.

2. **$$ \{3.50,7,10.50,14, \ldots, 42\} $$:**
   - **Incorrect.** This lists specific multiples of \$3.50, excluding other possible values.

3. **$$ \{p(g) \mid 0 \leq p(g) \leq 42\} $$:**
   - **Correct.** This accurately represents all possible costs from \$0 to \$42, inclusive.

4. **$$ \{p(g) \mid p(g) \leq 42\} $$:**
   - **Partially Correct but Incomplete.** While it includes all costs up to \$42, it doesn't specify the lower bound (which is \$0).

### **Conclusion**

The **range** of the function is best represented by:

**$$ \{p(g) \mid 0 \leq p(g) \leq 42\} $$**

This option accurately captures all possible cost values from \$0 up to \$42, considering the continuous nature of gas measurements.

**Answer:** 
$$\{p(g) \mid 0 \leq p(g) \leq 42\}$$
The range is all amounts from \$0 to \$42, inclusive.

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