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Let's break down the problem and solve it step by step.

### (a) Finding the linear function \( P(x) \)

The problem tells us that:
- In 2003, \( P(2003) = 38 \% \).
- In 2011, \( P(2011) = 58 \% \).

We want to find a linear function of the form:
\[
P(x) = mx + b
\]
Where:
- \( x \) is the year.
- \( P(x) \) is the percentage.
- \( m \) is the slope (rate of change).
- \( b \) is the y-intercept (initial percentage in 2003).

#### Step 1: Calculate the slope \( m \)
The slope \( m \) is given by the change in percentage divided by the change in years:
\[
m = \frac{P(2011) - P(2003)}{2011 - 2003}
\]
Substitute the values:
\[
m = \frac{58 - 38}{2011 - 2003} = \frac{20}{8} = 2.5
\]

#### Step 2: Find the y-intercept \( b \)
Now, use the point \( P(2003) = 38 \) to find \( b \). Substitute \( m = 2.5 \) and \( x = 2003 \) into the linear equation:
\[
P(2003) = 2.5(2003) + b = 38
\]
Solve for \( b \):
\[
2.5(2003) + b = 38
\]
\[
5007.5 + b = 38
\]
\[
b = 38 - 5007.5 = -4969.5
\]

Thus, the linear function is:
\[
P(x) = 2.5x - 4969.5
\]

### (b) Estimating when the percentage was between 43% and 53%

We need to solve for \( x \) when \( P(x) = 43 \% \) and \( P(x) = 53 \% \).

#### For \( P(x) = 43 \):
\[
43 = 2.5x - 4969.5
\]
Solve for \( x \):
\[
2.5x = 43 + 4969.5
\]
\[
2.5x = 5012.5
\]
\[
x = \frac{5012.5}{2.5} = 2005
\]

#### For \( P(x) = 53 \):
\[
53 = 2.5x - 4969.5
\]
Solve for \( x \):
\[
2.5x = 53 + 4969.5
\]
\[
2.5x = 5022.5
\]
\[
x = \frac{5022.5}{2.5} = 2009
\]

So, the percentage was between 43% and 53% between the years 2005 and 2009.

---

Would you like any more details or explanations? 

Here are 5 related questions:
1. How can we interpret the slope in the context of this problem?
2. What does the y-intercept represent in real-world terms here?
3. Can you predict what the percentage will be in 2020 using this model?
4. How would the equation change if the percentage in 2011 was different?
5. Can a linear model like this continue indefinitely, or are there limitations?

**Tip:** Linear functions are helpful in making predictions, but always consider whether the trend is sustainable over time!

Assume that in 2003 consumers used credit and debit cards to pay for 38% of all purchases. This percentage is projected to be 58% in 2011. (a) Find a linear function that models the data, where x is the year and P(x) is the percentage. (b) Estimate when this percentage was between 43% and 53%.

Learn how to find a linear function to model the percentage of credit and debit card usage from 2003 to 2011. Step-by-step solution includes calculating the slope and y-intercept, and estimating when the percentage was between 43% and 53%. Suitable for high school students.

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