f(x)= x + sin x.... find "f" inverse?

Question

x belongs to ( 0 , pi / 2 )

Answer 1

im going to prove that this function can't be inversed.
i mean i want to prove that this function is not an one-to-one function hence it does'nt have inverse:

y(1) = x(1) + sinx(1)
y(2) = x(2) + sinx(2)

y(1) = y(2) => x(1) + sinx(1) = x(2) + sinx(2) =>

x(1) - x(2) = sinx(2) - sinx(1) = 2sin( (x1- x2)/2 )cos( (x1 +x2)/2)

let A = x(1) - x(2)
=> 2sin( (x1- x2)/2 )cos( (x1 +x2)/2) = A
if A > 1 or A < -1 then this equation does'nt have any souloution
but is -1 < a < 1 then A = sinB
then 2sin( (x1- x2)/2 )cos( (x1 +x2)/2) = sinB = 2sinB/2cosB/2
so,this equation has "more than one" soloution,and hence,we can't just get x(1) = x(2) , so this function doesn't have inverse.

hope it helps!

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to sahsjing

dear sahsjing,you are right,of course it has,but we never say for example "arcsinx" is the inverse of sinx,we define the new Domain and then we say...

i graphed y = x + sinx and then x = y + siny,and it (almostly) made me sure that f^(-1)(x) is not a "function"...
thanks anyway my friend!

Answer 2

Go with Puggy on this one, no closed form (algebraic) answer, you only know that an inverse exists. You can approximate values, for instance, if you want to know f inverse at 1 you solve x+sin(x)=1 numerically and get the inverse value...........

Answer 3

You cannot find an explicit inverse function of f(x). However, you can graph it.

First, graph f(x) = x + sin x on a thin paper, then switch x and y axises. Viewing from the back of the paper, you can see a perfect f^-1(x).

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farbod f,

A function can always have its inverse in its monotonic domain. You can think about sin x, cos x etc.

Answer 4

Normally, when solving an inverse you set f(x) to y,

y = x + sinx

And then you switch the x and y variables and try to solve for y

x = y + siny

However, it isn't possible to solve for y using elementary methods. Therefore, the inverse itself isn't easy to solve for. All we can determine are properties of the curve siny + y = x (using Calculus).

Answer 5

let us take f(x)=x then itsinverse is x itself
now consider f(x)=sinxthen its inverse is sin-(x)
hence the inverse of ur function is f-(x)=x+sin-(x)

Answer 6

I don't know how to solve it but you switch the x's with y's and change f(x) to x to make x = y + sin y, but I'm sure you already know to do that.