https://r2.erweima.ai/mathbot/image/1726616931229w9655aj2.jpg

The given formula in the image is:

\[
C = \frac{5}{9}(F - 32)
\]

where \(F \geq -459.67\) represents the temperature in Fahrenheit, and \(C\) is the temperature in Celsius. The task is to find the inverse function, which expresses \(F\) as a function of \(C\), and determine the domain of this inverse function.

### Step 1: Solve for \( F \) (Inverse Function)
Starting with the original equation:

\[
C = \frac{5}{9}(F - 32)
\]

Multiply both sides by \( \frac{9}{5} \) to eliminate the fraction:

\[
\frac{9}{5}C = F - 32
\]

Now, add 32 to both sides to solve for \( F \):

\[
F = \frac{9}{5}C + 32
\]

Thus, the inverse function is:

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

### Step 2: Determine the Domain of the Inverse Function
In the original formula, we are given that \( F \geq -459.67 \), which is the absolute zero temperature in Fahrenheit. To find the domain of the inverse function, we substitute \( F = -459.67 \) into the inverse function to find the corresponding temperature in Celsius:

\[
C = \frac{5}{9}(-459.67 - 32) = \frac{5}{9}(-491.67) \approx -273.15
\]

Therefore, the domain of the inverse function is \( C \geq -273.15 \), representing the absolute zero temperature in Celsius.

### Interpretation
The inverse function allows us to convert a temperature from Celsius to Fahrenheit, and its domain is all temperatures in Celsius greater than or equal to -273.15°C (absolute zero).

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Would you like more details or have any questions about this process?

Here are 5 related questions to deepen your understanding:
1. What is the range of the inverse function in this context?
2. How would you graph the original and inverse functions?
3. Can you explain why absolute zero in Celsius and Fahrenheit are different values?
4. How can this formula be applied in real-world contexts involving temperature conversion?
5. What are the steps to verify that the functions are truly inverses of each other?

**Tip:** When dealing with inverse functions, remember to swap the roles of the dependent and independent variables carefully and check for any restrictions in the domain and range.

A fórmula C = (5/9)(F - 32), onde F ≥ -459,67 expressa a temperatura 'C' em graus Celsius como função da temperatura na escala da temperatura na escala 'F', em Fahrenheit. Encontre uma fórmula para a função inversa e a interprete. Qual o domínio dessa função inversa?

Learn how to find the inverse function of the Celsius to Fahrenheit conversion formula. This problem involves algebraic manipulation and covers key concepts such as inverse functions and temperature conversion. Suitable for high school students.

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