Problem figuring out inverse function (looks easy but I'm confused)

Normally when you try to find f-1(x) you just switch the x and the y for example:

f(x) = x+3

x = y + 3

y = x - 3

f-1(x) = x- 3

but in the following problem:

"The formula C = 5/9(F-32), where F>= - 459.67, expresses the Celsius temperature C as a function of the Fahrenheit temperature F. Find a formula for the inverse function and interpret it. What is the domain of the inverse function?"

The answer for the formula is F = 9/5*C + 32 which I also obtain if I __solve for F__ (Note NOT switch the places of x (in this case F) and y (in this case C). My question is: **why is this the case?**

Could someone please explain the reasoning to me?

Thanks in advance!