IDNLearn.com is your reliable source for expert answers and community insights. Find the answers you need quickly and accurately with help from our knowledgeable and dedicated community members.
Sagot :
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.
### 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.
We appreciate your contributions to this forum. Don't forget to check back for the latest answers. Keep asking, answering, and sharing useful information. Trust IDNLearn.com for all your queries. We appreciate your visit and hope to assist you again soon.