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!
Wait something is wrong my example:
you said that I'm right but according to your explanation, I am wrong also in my example.
I got f-1(x) = x - 3 = y
and you said x = y - 3 --> f-1(x) = x - 3 but if you rearrange what you wrote in bold, you'd get y = x+3 and not y = x-3.
I did switch in my example because:
f(x) = y and f-1(x) = y (of inverse function)
y = x + 3
x = y + 3
y = x - 3
f-1(x) = x - 3
Something is either wrong or really confusing here so I was hoping I could get this cleared before proceeding to the actual problem.
Anyway, and in case case of doubt, just compose the original function with the one you thinbk is its inverse: if you get the identity function you're fine, otherwise you're not.
Tonio, you can do it either way and most people are taught to swap variable first.:
"Late swapping", f(x)= y= x+ 3 so y- 3= x. Now swap: .
"Early swapping", f(x)= y= x+ 3 so swapping variables, x= y+ 3. Now solve for y: x- 3= y so .
I prefer "early swapping" because I think the crucial point to make is that going from f to f inverse does swap domain and range. The rest is just manipulation!
The "Fahrenheit", "Celcius" is just an annoying abberation! If you had f(x)= 5/9(x-32) and were asked to find the inverse function, you would say "y= (5/9)(x- 32) so I first swap x and y: x= (5/9)(y- 32) and solve for y: (9/5)x= y- 32 so y= (9/5)x+ 32. .
However, here, the variables have specific physical meaning. In the first x was temperature in degrees Fahrenheit, which we call "F" to remind us of that and the result of the function is the temperature in degrees Celcius which we call "C".
Now, the inverse function is = (9/5)x+ 32 but now we are going the other way (that's what "inverse" does) so to remind ourselves of this, we Label " as "F" and x as "C".
F= (9/5)C+ 32.