How do we approximate #f^(-1)(x)#?

If#f(x)=x+sin(xpi)#, how could we go about trying to find its inverse over an interval on which #f# is invertible?

1 Answer
Steve M
Nov 17, 2017

We have:

# f(x)=x+sin(pix) #

If we let:

# y = x+sin(pix) # ..... [A]

Then, the inverse, #f^(-1)(x)# would normally be obtained by re-arranging equation [A] into the form:

# x = g(y) #

Making #g# the inverse.

However we cannot perform such a re-arrangement using the standard elementary functions that we are familiar with.

This is where the "approximation" component comes in because given any particular constant value of #x#, (let us call this value #alpha#, say) we would find the inverse #f^(-1)(alpha)# by solving numerically the equation:

# x+sin(pix) =alpha #

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