To solve the problem, let's proceed step-by-step:
1. Understanding the Functions:
- We have the function [tex]\( C(F) = \frac{5}{9}(F - 32) \)[/tex] which converts a temperature from Fahrenheit to Celsius.
- We also have the function [tex]\( K(C) = C + 273.15 \)[/tex] which converts a temperature from Celsius to Kelvin.
2. Composing the Functions:
- To convert directly from Fahrenheit to Kelvin, we need to substitute the Celsius function into the Kelvin function.
Therefore:
- [tex]\( K(F) = K(C(F)) \)[/tex]
- Substituting [tex]\( C(F) \)[/tex] into [tex]\( K \)[/tex]:
[tex]\[
K(F) = K\left(\frac{5}{9}(F - 32)\right)
\][/tex]
- This gives us:
[tex]\[
K(F) = \frac{5}{9}(F - 32) + 273.15
\][/tex]
3. Calculating the Temperature in Kelvin:
- Now, we substitute [tex]\( F = 77 \)[/tex] into our composed function [tex]\( K(F) \)[/tex]:
[tex]\[
K(77) = \frac{5}{9}(77 - 32) + 273.15
\][/tex]
- Evaluating this expression, we obtain:
[tex]\[
K(77) = 25.0 + 273.15 = 298.15
\][/tex]
In summary:
1. The composed function to convert Fahrenheit directly to Kelvin is:
[tex]\[
\boxed{\frac{5}{9}(F - 32) + 273.15}
\][/tex]
2. Using this function, [tex]\( 77^\circ F \)[/tex] is converted to:
[tex]\[
\boxed{298.15 \text{ K}}
\][/tex]
