Questions: The function defined by F(x) = 9/5 x + 32 gives the temperature F(x) (in degrees Fahrenheit) based on the temperature x (in Celsius). (a) Determine the temperature in Fahrenheit if the temperature in Celsius is 10°C. (b) Write a function representing the inverse of F and interpret its meaning in context. (c) Determine the temperature in Celsius if the temperature in Fahrenheit is 23°F.

Solution Steps

To find the temperature in Fahrenheit when the temperature in Celsius is \(10^{\circ} \mathrm{C}\), we use the function:

\[ F(x) = \frac{9}{5} x + 32 \]

Substitute \(x = 10\):

\[ F(10) = \frac{9}{5} \times 10 + 32 = 18 + 32 = 50 \]

To find the inverse of the function \(F(x) = \frac{9}{5} x + 32\), we need to solve for \(x\) in terms of \(F(x)\).

1. Start with the equation: \[ y = \frac{9}{5} x + 32 \]

2. Swap \(x\) and \(y\) to find the inverse: \[ x = \frac{9}{5} y + 32 \]

3. Solve for \(y\): \[ x - 32 = \frac{9}{5} y \] \[ y = \frac{5}{9} (x - 32) \]

The inverse function is: \[ F^{-1}(x) = \frac{5}{9} (x - 32) \]

This inverse function converts a temperature from Fahrenheit back to Celsius.

To find the temperature in Celsius when the temperature in Fahrenheit is \(23^{\circ} \mathrm{F}\), use the inverse function:

\[ F^{-1}(x) = \frac{5}{9} (x - 32) \]

Substitute \(x = 23\):

\[ F^{-1}(23) = \frac{5}{9} (23 - 32) = \frac{5}{9} \times (-9) = -5 \]

Final Answer