Sure, let's go through the problem step-by-step.
### Part A:
We are asked to find the inverse function [tex]\( F^{-1}(x) \)[/tex] of the function [tex]\( F(x) = \frac{9}{5}x + 32 \)[/tex].
To find the inverse function, we follow these steps:
1. Set [tex]\( y = F(x) \)[/tex]:
[tex]\[ y = \frac{9}{5}x + 32 \][/tex]
2. Solve for [tex]\( x \)[/tex]:
- Subtract 32 from both sides to isolate the term with [tex]\( x \)[/tex]:
[tex]\[ y - 32 = \frac{9}{5}x \][/tex]
- Multiply both sides by [tex]\( \frac{5}{9} \)[/tex] to solve for [tex]\( x \)[/tex]:
[tex]\[ x = \frac{5}{9}(y - 32) \][/tex]
3. Replace [tex]\( y \)[/tex] with [tex]\( x \)[/tex] to express the inverse function:
[tex]\[ F^{-1}(x) = \frac{5}{9}(x - 32) \][/tex]
Therefore, the correct expression that models [tex]\( F^{-1}(x) \)[/tex] is:
[tex]\[ \boxed{\frac{5}{9}(x - 32)} \][/tex]
This corresponds to option 4 from the given choices.
### Part B:
Next, we need to evaluate the inverse function at [tex]\( x = 50 \)[/tex]. That is, we need to find the value of [tex]\( F^{-1}(50) \)[/tex].
Using the inverse function we found:
[tex]\[ F^{-1}(x) = \frac{5}{9}(x - 32) \][/tex]
Substitute [tex]\( x = 50 \)[/tex]:
[tex]\[ F^{-1}(50) = \frac{5}{9}(50 - 32) \][/tex]
Perform the calculation inside the parentheses first:
[tex]\[ 50 - 32 = 18 \][/tex]
Then multiply by [tex]\( \frac{5}{9} \)[/tex]:
[tex]\[ F^{-1}(50) = \frac{5}{9} \times 18 = 10 \][/tex]
The value of [tex]\( F^{-1}(50) \)[/tex] is:
[tex]\[ \boxed{10} \][/tex]
Interpretation:
This value represents the temperature in degrees Celsius when the temperature is 50 degrees Fahrenheit. So, [tex]\( F^{-1}(50) = 10 \)[/tex] means that 50 degrees Fahrenheit is equivalent to 10 degrees Celsius.
To summarize:
Part A: The inverse function is [tex]\( \frac{5}{9}(x - 32) \)[/tex].
Part B: The value of [tex]\( F^{-1}(50) \)[/tex] is 10, meaning 50 degrees Fahrenheit is equal to 10 degrees Celsius.
