Find formula for inverse function of C = 5/9 * (F - 32)

[MarkSA]

Hello,

I'm a little confused about finding inverse functions. The way that I understood it, and the steps the book illustrates, are as follows:
1) swap x's and y's
2) solve for y
3) result is the inverse function

This seemed to work for most functions I was given... However there was a word problem that said:
1) The formula C = 5/9(F - 32) expresses Celsius as a function of Fahrenheit. Find a formula for the inverse function.
Well.. C as a function of F is like saying y(x), so I let C = y and F = x. Then I end up with:
Find the inverse function of y = 5/9 * (x - 32)

So I swap the variables..
x = 5/9 * (y - 32)

Solve for y:
9/5 * x = y - 32
y = (9/5 * x) + 32

And since y is C and F is x,
C = (9/5 * F) + 32

But that isn't the right answer. The right answer is F = (9/5 * C) + 32.

Can you clarify where I am going wrong on this? I've had the same issue in other word problems. How to know when to swap the variables or when not to? (it appears they were not swapped in this problem, merely solved for F). Thanks

 

[stapel]

So you didn't actually understand the reply you received on your other thread on this question...?

Note: When you find the inverse function, the x and y are swapped: whatever was C is now F, and whatever was F is now C. Or, at least, they should be.

Try doing the exercise without going to the x and y; keep the equation in terms of F and C, like you were given before. Then maybe you won't trip yourself up.

Eliz.

 

[MarkSA]

Yes I first looked at the other thread and thought I understood it but I misread the steps.

I had another word problem today that dealt with the same issue and I was running into the same trouble as before...

If I solve it in terms of F and C, to get the right answer I would have to just solve for F.. but to do that, there is no swapping involved, right? I thought it was the swapping bit that actually made a function an inverse function to the original?

Because If i'm doing a regular problem like Find a formula for the inverse of the function:
y = 3 - 2x
just solving for x doesn't give me the correct answer. I would get x = -(y - 3)/2 but the correct answer appears to be y = -(x - 3)/2

 

[pka]

Yes I first looked at the other thread and thought I understood it but I misread the steps.
I thought it was the swapping bit that actually made a function an inverse function to the original?

I think that you are a victim of some abuse of the language concerning ‘inverse’.
Consider this, if \(\displaystyle C = f(F)\quad \Rightarrow \quad f^{ - 1} (C) = F\).
So we get
\(\displaystyle \begin{array}{l} C = f(F) = \frac{5}{9}\left( {F - 32} \right) \\ f^{ - 1} \left( F \right) = \frac{{9F + 160}}{5}\quad \Rightarrow \quad F = f^{ - 1} \left( C \right) = \frac{{9C + 160}}{5} \\ \end{array}\).