Let's go through this problem step-by-step.
### Part A
We need to find the inverse function [tex]\( F^{-1}(x) \)[/tex]. We're given the function:
[tex]\[ F(x) = \frac{9}{5}x + 32 \][/tex]
To find the inverse function [tex]\( F^{-1}(x) \)[/tex], we will perform the following steps:
1. Start with the equation [tex]\( y = F(x) \)[/tex]:
[tex]\[
y = \frac{9}{5}x + 32
\][/tex]
2. Swap the variables [tex]\( y \)[/tex] and [tex]\( x \)[/tex]:
[tex]\[
x = \frac{9}{5}y + 32
\][/tex]
3. Solve for [tex]\( y \)[/tex]:
[tex]\[
x - 32 = \frac{9}{5}y
\][/tex]
[tex]\[
y = \frac{5}{9}(x - 32)
\][/tex]
So the inverse function [tex]\( F^{-1}(x) \)[/tex] is:
[tex]\[ F^{-1}(x) = \frac{5}{9}(x - 32) \][/tex]
Comparing with the given choices, the correct expression that models [tex]\( F^{-1}(x) \)[/tex] is:
[tex]\[ \frac{5}{9}(x-32) \][/tex]
### Part B
Next, we need to determine the value of [tex]\( F^{-1}(50) \)[/tex] and interpret its significance.
Plugging [tex]\( x = 50 \)[/tex] into the inverse function:
[tex]\[
F^{-1}(50) = \frac{5}{9}(50 - 32)
\][/tex]
We already know from the result:
[tex]\[ F^{-1}(50) = 10.0 \][/tex]
So, the value of [tex]\( F^{-1}(50) \)[/tex] is [tex]\( 10.0 \)[/tex].
This value represents the temperature in degrees Celsius when the temperature is 50 degrees Fahrenheit.
To complete the sentence:
The value is 10.0 and it represents the temperature in degrees Celsius when the temperature is 50 degrees Fahrenheit.
