The relationship between the percentage of frozen citrus crop, and the cost of box of oranges is an illustration of a linear function.
- <em>The linear equation of the function is: </em>
<em>, while its inverse is: </em>
<em>.</em> - <em>The domain is from 0% to 100%, while the range is from 7 to 29.9</em>
<em />
Given that:
![c = g(P)](https://tex.z-dn.net/?f=c%20%3D%20g%28P%29)
<u>Input and Output quantity</u>
The input quantity is the <em>percentage of frozen citrus crop</em>, while the output quantity is the <em>cost of box of oranges</em>
<u>Linear Function</u>
The given parameters can be written as:
![(P_1,c_1) = (20\%, 11.58)](https://tex.z-dn.net/?f=%28P_1%2Cc_1%29%20%3D%20%2820%5C%25%2C%2011.58%29)
![(P_2,c_2) = (80\%, 25.32)](https://tex.z-dn.net/?f=%28P_2%2Cc_2%29%20%3D%20%2880%5C%25%2C%2025.32%29)
Calculate the slope (m)
![m = \frac{c_2 - c_1}{P_2 - P_1}](https://tex.z-dn.net/?f=m%20%3D%20%5Cfrac%7Bc_2%20-%20c_1%7D%7BP_2%20-%20P_1%7D)
So, we have:
![m = \frac{25.32 - 11.58}{80\% - 20\%}](https://tex.z-dn.net/?f=m%20%3D%20%5Cfrac%7B25.32%20-%2011.58%7D%7B80%5C%25%20-%2020%5C%25%7D)
![m = \frac{13.74}{60\%}](https://tex.z-dn.net/?f=m%20%3D%20%5Cfrac%7B13.74%7D%7B60%5C%25%7D)
![m = 22.9](https://tex.z-dn.net/?f=m%20%3D%2022.9)
The equation is then calculated using:
![c =m(P - p_1) + c_1](https://tex.z-dn.net/?f=c%20%3Dm%28P%20-%20p_1%29%20%20%2B%20c_1)
So, we have:
![c =22.9(P - 20\%) + 11.58](https://tex.z-dn.net/?f=c%20%3D22.9%28P%20-%2020%5C%25%29%20%20%2B%2011.58)
![c =22.9P - 4.58 + 11.58](https://tex.z-dn.net/?f=c%20%3D22.9P%20-%204.58%20%20%2B%2011.58)
![c =22.9P +7](https://tex.z-dn.net/?f=c%20%3D22.9P%20%2B7)
So, the function is:
![g(P) =22.9P +7](https://tex.z-dn.net/?f=g%28P%29%20%3D22.9P%20%2B7)
<u>The domain and the range</u>
The domain is the <em>possible input value (i.e. possible values of P).</em>
Because P is a percentage, its possible values are 0% to 100%.
Hence, the domain of the function is: ![[0\%,100\%]](https://tex.z-dn.net/?f=%5B0%5C%25%2C100%5C%25%5D)
The range is the <em>possible output value (i.e. possible values of c).</em>
When P = 0% and 100%
![c = 22.9 \times 0\% + 7 = 7](https://tex.z-dn.net/?f=c%20%3D%2022.9%20%5Ctimes%200%5C%25%20%2B%207%20%3D%207)
![c = 22.9 \times 100\% + 7 = 29.9](https://tex.z-dn.net/?f=c%20%3D%2022.9%20%5Ctimes%20100%5C%25%20%2B%207%20%3D%2029.9)
Hence, the range of the function is: ![[7,29.9]](https://tex.z-dn.net/?f=%5B7%2C29.9%5D)
<u>The meaning of </u>
<u />
is an inverse equation, where 12 represents the <em>cost of box of oranges.</em>
So,
represents the percentage of frozen citrus crop, when the cost is $12.
<u>The inverse formula</u>
We have:
![c =22.9P +7](https://tex.z-dn.net/?f=c%20%3D22.9P%20%2B7)
Make P the subject
![22.9P = c - 7](https://tex.z-dn.net/?f=22.9P%20%3D%20c%20-%207)
Divide by 22.9
![P = \frac{c - 7}{22.9}](https://tex.z-dn.net/?f=P%20%3D%20%5Cfrac%7Bc%20-%207%7D%7B22.9%7D)
So, the inverse function is:
![g^{-1}(c) = \frac{c - 7}{22.9}](https://tex.z-dn.net/?f=g%5E%7B-1%7D%28c%29%20%3D%20%5Cfrac%7Bc%20-%207%7D%7B22.9%7D)
<em>This is used to calculate the percentage of frozen citrus crop, when the cost is known.</em>
<em />
![g^{-1}(c) = \frac{c - 7}{22.9}](https://tex.z-dn.net/?f=g%5E%7B-1%7D%28c%29%20%3D%20%5Cfrac%7Bc%20-%207%7D%7B22.9%7D)
Substitute 12 for c
![g^{-1}(12) = \frac{12 - 7}{22.9}](https://tex.z-dn.net/?f=g%5E%7B-1%7D%2812%29%20%3D%20%5Cfrac%7B12%20-%207%7D%7B22.9%7D)
![g^{-1}(12) = \frac{5}{22.9}](https://tex.z-dn.net/?f=g%5E%7B-1%7D%2812%29%20%3D%20%5Cfrac%7B5%7D%7B22.9%7D)
![g^{-1}(12) = 22\%](https://tex.z-dn.net/?f=g%5E%7B-1%7D%2812%29%20%3D%2022%5C%25)
Read more about linear equations at:
brainly.com/question/19770987