The function f(c)=9/5c+32 allows you to convert degrees celsius to degrees fahrenheit. Find the inverse of the function so that you can convert degrees fahrenheit back to degrees celsius.
The Correct Answer and Explanation is:
To find the inverse of the function ( f(c) = \frac{9}{5}c + 32 ), which converts degrees Celsius (C) to degrees Fahrenheit (F), we need to express ( c ) in terms of ( f ). The inverse function will allow us to convert degrees Fahrenheit back to degrees Celsius.
Step 1: Replace ( f(c) ) with ( F )
Start by rewriting the equation as follows:
[
F = \frac{9}{5}c + 32
]
Step 2: Isolate ( c )
To isolate ( c ), we first subtract 32 from both sides:
[
F – 32 = \frac{9}{5}c
]
Next, multiply both sides by ( \frac{5}{9} ) to solve for ( c ):
[
c = \frac{5}{9}(F – 32)
]
Step 3: Write the Inverse Function
Thus, the inverse function can be expressed as:
[
f^{-1}(F) = \frac{5}{9}(F – 32)
]
Explanation
The original function ( f(c) = \frac{9}{5}c + 32 ) is a linear transformation that takes a temperature in Celsius and converts it to Fahrenheit. The formula is derived from the relationship between the two temperature scales, where 32 degrees Fahrenheit corresponds to 0 degrees Celsius. The factor ( \frac{9}{5} ) represents the ratio of the degree sizes in the two scales.
When we find the inverse function, we essentially “reverse” this process. By substituting ( F ) for ( f(c) ), we can rearrange the equation to solve for ( c ). This allows us to determine the Celsius equivalent of any given Fahrenheit temperature.
The function ( f^{-1}(F) = \frac{5}{9}(F – 32) ) indicates that to convert Fahrenheit back to Celsius, we first subtract 32 from the Fahrenheit value and then multiply the result by ( \frac{5}{9} ). This operation correctly accounts for the difference in scale between the two temperature measurements, ensuring an accurate conversion. Thus, the inverse function serves a crucial role in thermodynamic calculations where temperature conversion is necessary.