Let's address each part of the question step-by-step:
### Part (a) Total Spending in 2006 and 2012
The given function for spending on pets [tex]\( P(x) \)[/tex] in billions of dollars is:
[tex]\[ P(x) = 2.1781x + 25.2 \][/tex]
Here, [tex]\( x \)[/tex] represents the number of years after the year 2000.
#### Calculating for the year 2006:
First, we need to find [tex]\( x \)[/tex] for the year 2006:
[tex]\[ x = 2006 - 2000 = 6 \][/tex]
So, to find the total spending in 2006:
[tex]\[ P(6) = 2.1781 \cdot 6 + 25.2 \][/tex]
After substituting and calculating, we find:
[tex]\[ P(6) = 38.2686 \][/tex]
Thus, the total amount of spending on pets in 2006 was [tex]\(\$38.2686\)[/tex] billion.
#### Calculating for the year 2012:
Next, we need to find [tex]\( x \)[/tex] for the year 2012:
[tex]\[ x = 2012 - 2000 = 12 \][/tex]
So, to find the total spending in 2012:
[tex]\[ P(12) = 2.1781 \cdot 12 + 25.2 \][/tex]
After substituting and calculating, we get:
[tex]\[ P(12) = 51.3372 \][/tex]
Thus, the total amount of spending on pets in 2012 was [tex]\(\$51.3372\)[/tex] billion.
### Part (b) The Inverse Function and Its Meaning
To find the inverse of [tex]\( P(x) \)[/tex], which we denote as [tex]\( P^{-1}(x) \)[/tex], we need to solve for [tex]\( x \)[/tex] in terms of [tex]\( P(x) \)[/tex].
Starting with the equation:
[tex]\[ P(x) = 2.1781x + 25.2 \][/tex]
1. Set [tex]\( P(x) = y \)[/tex]:
[tex]\[ y = 2.1781x + 25.2 \][/tex]
2. Solve for [tex]\( x \)[/tex]:
[tex]\[ y - 25.2 = 2.1781x \][/tex]
[tex]\[ x = \frac{y - 25.2}{2.1781} \][/tex]
Therefore, the inverse function [tex]\( P^{-1}(y) \)[/tex] is:
[tex]\[ P^{-1}(y) = \frac{y - 25.2}{2.1781} \][/tex]
#### Interpretation:
The inverse function [tex]\( P^{-1}(y) \)[/tex] gives the value of [tex]\( x \)[/tex] (the number of years after 2000) when the total spending on pets reaches [tex]\( y \)[/tex] billion dollars. Essentially, [tex]\( P^{-1}(y) \)[/tex] enables us to find out in which year a certain amount of spending on pets occurred. For example, if we know the total spending is \$40 billion, using [tex]\( P^{-1} \)[/tex], we can find the exact year corresponding to that spending.
