I need help with this quetziony

A devastating freeze in California’s Central Valley in January 2007 wiped out approximately 75% of the state’s citrus crop. It t

inna [77]

The relationship between the percentage of frozen citrus crop, and the cost of box of oranges is an illustration of a linear function.

The linear equation of the function is: $g(P) =22.9P +7$ , while its inverse is: $g^{-1}(c) = \frac{c - 7}{22.9}$ .
The domain is from 0% to 100%, while the range is from 7 to 29.9

Given that:

$c = g(P)$

Input and Output quantity

The input quantity is the percentage of frozen citrus crop, while the output quantity is the cost of box of oranges

Linear Function

The given parameters can be written as:

$(P_1,c_1) = (20\%, 11.58)$

$(P_2,c_2) = (80\%, 25.32)$

Calculate the slope (m)

$m = \frac{c_2 - c_1}{P_2 - P_1}$

So, we have:

$m = \frac{25.32 - 11.58}{80\% - 20\%}$

$m = \frac{13.74}{60\%}$

$m = 22.9$

The equation is then calculated using:

$c =m(P - p_1) + c_1$

So, we have:

$c =22.9(P - 20\%) + 11.58$

$c =22.9P - 4.58 + 11.58$

$c =22.9P +7$

So, the function is:

$g(P) =22.9P +7$

The domain and the range

The domain is the possible input value (i.e. possible values of P).

Because P is a percentage, its possible values are 0% to 100%.

Hence, the domain of the function is: $[0\%,100\%]$

The range is the possible output value (i.e. possible values of c).

When P = 0% and 100%

$c = 22.9 \times 0\% + 7 = 7$

$c = 22.9 \times 100\% + 7 = 29.9$

Hence, the range of the function is: $[7,29.9]$

The meaning of $g^{-1}(12)$

$g^{-1}(12)$ is an inverse equation, where 12 represents the cost of box of oranges.

So, $g^{-1}(12)$ represents the percentage of frozen citrus crop, when the cost is $12.

The inverse formula

We have:

$c =22.9P +7$

Make P the subject

$22.9P = c - 7$

Divide by 22.9

$P = \frac{c - 7}{22.9}$

So, the inverse function is:

$g^{-1}(c) = \frac{c - 7}{22.9}$

This is used to calculate the percentage of frozen citrus crop, when the cost is known.

$g^{-1}(c) = \frac{c - 7}{22.9}$

Substitute 12 for c

$g^{-1}(12) = \frac{12 - 7}{22.9}$

$g^{-1}(12) = \frac{5}{22.9}$

$g^{-1}(12) = 22\%$

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