Help with inverse function problem

[crm08]

Homework Statement

If f(x) = x + cos(x), find f-inverse of (1)

Homework Equations

**first time asking a forum question, please inform me of any errors in posting this question

The Attempt at a Solution

(1) y = x + cos(x) => if y = f(x)

(2) x = y + cos(y) => if x = f(y)

(3) ** solve for y? this is where I am stuck, is it possible to completely show work and solve for "y" before plugging in "1"? I tried switching the variables and using the solve command on my ti-89 and it gives the answer:
cos(y) + y = x

 

[jgens]

I don't think there is anyway of expressing the inverse of that function in terms of elementary function (someone with a more authoritative math background could clarify). However, if we note the definition of an inverse function: f:X -> Y than f^-1:Y -> X we can find a solution for f^-1(1). Hence, to find f^-1(1) all we need find is the x value which produces an f(x) value of 1; therefore, 1 = x + cos(x). Can you solve that?

 

[crm08]

Ok gotcha, x = 0, thanks for the help

 

[jgens]

Yep, so f^-1(1) = 0.

 

[mutton]

There is no other solution to f(x) = 1 because f is nondecreasing everywhere and increasing at 0 (use f' to show this).