How can you tell if this function if invertible? And if so, how would you find the inverse function?

Question

The function F = g(C) = 32 + 9/5C converts a temperature in degrees Celsius to degrees Fahrenheit.
( a ) Explain how you can tell that this function is invertible.
( b ) If it invertible, find the inverse function C= g^(-1) (F)

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Answer 1 · 2017-11-17T08:25:25

#C = g^(-1)(F) = 5/9(F-32)#

Given:

#F = 32+9/5 C#

Subtract #32# from both sides to get:

#F-32 = 9/5 C#

Multiply both sides by #5/9# and transpose to get:

#C = 5/9(F-32)#

Note that both of these steps are reversible, so the equation we have arrived at is exactly equivalent to the original equation.

The derived equation uniquely determines #C# for any given value of #F#, so defines a function, which is the inverse function:

#g^(-1)(F) = 5/9(F-32)#

Since we have found an inverse function, we have completed b) and proved a).