Sure, let's go through each part of the problem step-by-step.
### Part (a)
We start with the function \( F(x) = \frac{9}{5}x + 32 \), which converts a temperature given in Celsius (\( x \)) to Fahrenheit (\( F(x) \)).
To determine the temperature in Fahrenheit when the temperature in Celsius is \( 25^\circ C \), we substitute \( x = 25 \) into the function:
[tex]\[
F(25) = \frac{9}{5} \cdot 25 + 32
\][/tex]
Performing the arithmetic:
[tex]\[
F(25) = \frac{9}{5} \cdot 25 + 32 = 45 + 32 = 77.0
\][/tex]
Therefore, the temperature in Fahrenheit is \( 77.0^\circ F \).
### Part (b)
To find the inverse function of \( F \), denoted as \( F^{-1} \), we need to express \( x \) in terms of \( F \). The inverse function will convert a temperature given in Fahrenheit back to Celsius.
Starting with the equation:
[tex]\[
F = \frac{9}{5}x + 32
\][/tex]
We will solve for \( x \) in terms of \( F \):
[tex]\[
F - 32 = \frac{9}{5}x
\][/tex]
Next, multiply both sides by \( \frac{5}{9} \):
[tex]\[
x = \frac{5}{9}(F - 32)
\][/tex]
Thus, the inverse function is:
[tex]\[
F^{-1}(F) = \frac{5}{9}F - \frac{160}{9}
\][/tex]
In a more simplified form, it can be written as:
[tex]\[
F^{-1}(F) = 0.555555555555556F - 17.7777777777778
\][/tex]
The meaning of this inverse function in context is that it converts the temperature from Fahrenheit back to Celsius.
### Part (c)
We need to determine the temperature in Celsius if the temperature in Fahrenheit is \( 23^\circ F \). Using the inverse function \( F^{-1}(F) = 0.555555555555556F - 17.7777777777778 \), we substitute \( F = 23 \):
[tex]\[
x = 0.555555555555556 \cdot 23 - 17.7777777777778
\][/tex]
Performing the arithmetic:
[tex]\[
x = 12.7777777777778 - 17.7777777777778 = -5.0
\][/tex]
Therefore, the temperature in Celsius is [tex]\( -5.0^\circ C \)[/tex].
