Finding the Inverse of a Function
The formula for converting a temperature in degrees Celsius to degrees Fahrenheit is [tex]F(C)=\frac{9}{5} C+32[/tex]. The inverse function will convert a temperature in degrees Fahrenheit to degrees Celsius. What is the formula for converting from degrees Fahrenheit to degrees Celsius?
[tex]C(F)=\square F-\square[/tex].

Start with the Celsius to Fahrenheit conversion formula: F = 5 9 ​ C + 32 .
Swap F and C and solve for C : F − 32 = 5 9 ​ C .
Multiply both sides by 9 5 ​ : C = 9 5 ​ ( F − 32 ) .
Distribute and simplify to get the final formula: C ( F ) = 9 5 ​ F − 9 160 ​ .

Explanation

Understanding the Problem We are given the formula to convert Celsius to Fahrenheit: F ( C ) = 5 9 ​ C + 32 . We need to find the inverse function C ( F ) that converts Fahrenheit to Celsius.

Setting up the Equation To find the inverse function, we will replace F ( C ) with F and solve for C in terms of F . So, we have: F = 5 9 ​ C + 32

Isolating the Term with C First, subtract 32 from both sides of the equation: F − 32 = 5 9 ​ C

Solving for C Next, multiply both sides of the equation by 9 5 ​ to isolate C : 9 5 ​ ( F − 32 ) = C

Rewriting as the Inverse Function Now, we can rewrite this as the inverse function C ( F ) : C ( F ) = 9 5 ​ F − 9 5 ​ ( 32 ) C ( F ) = 9 5 ​ F − 9 160 ​

Final Formula So, the formula for converting Fahrenheit to Celsius is: C ( F ) = 9 5 ​ F − 9 160 ​


Examples
Imagine you're a chef who needs to convert oven temperatures quickly between Fahrenheit and Celsius. Knowing how to convert between these scales allows you to accurately follow recipes from different parts of the world. This conversion is also crucial in scientific experiments where precise temperature measurements are essential. Understanding inverse functions helps in many practical scenarios where reversing a process or formula is necessary.

Answered by GinnyAnswer | 2025-07-08

To find the inverse function for converting Fahrenheit to Celsius, we start from the formula F ( C ) = 5 9 ​ C + 32 . By isolating C , we derive the formula C ( F ) = 9 5 ​ F − 9 160 ​ . This formula converts temperatures from Fahrenheit back to Celsius accurately.
;

Answered by Anonymous | 2025-07-16