The function defined by [tex]F(x)=\frac{9}{5} x+32[/tex] gives the temperature [tex]F(x)[/tex] (in degrees Fahrenheit) based on the temperature [tex]x[/tex] (in Celsius).

(a) Determine the temperature in Fahrenheit if the temperature in Celsius is [tex]25^{\circ} C[/tex].

(b) Write a function representing the inverse of [tex]F[/tex] and interpret its meaning in context.

(c) Determine the temperature in Celsius if the temperature in Fahrenheit is [tex]23^{\circ} F[/tex].



Answer :

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].