# Given #f(x) = x^3-3x#, how can you construct an infinitely differentiable one-one function #g(x):RR->RR# with #g(x) = f(x)# in #(-oo, -2] uu [2, oo)#?

##### 4 Answers

Here is my first attempt. (Unfortunately, it fails.)

#### Explanation:

graph{x^3-3x [-5.454, 5.645, -2.606, 2.94]}

My first thought is for a piecewise function, keeping the definition of

I tried a piece based on

We need

and

Determined to find a polynomial, I set

#g(x)=ax^n# where#n# is odd

So,

We need

This lead to

But for infinite differentiability, we also need

#g''(2) =-12# and# g''(2) = 12#

#g'''(-2) = g'''(2) = 6#

#g^((n))(-2) = g^((n))(2) = 0# for#n>= 4# .

This attempt fails to satisfy those.

My initial thought was to try something using

Perhaps we can start with and odd, four-times integrable function and work backwards from there.

So

#### Explanation:

This makes

(All integrals by Wolfram Alpha) And

(Note:

#g''(+-2) = 3/4(+-2)(8) = +-12# )

Never mind.

What was I thinking?

See below.

#### Explanation:

For practical purposes and with a guarantee of accuracy, we can proceed, according to

This for

Solving this system we obtain

So the binding function is

Attached a plot showing the binding.

#### Explanation:

A classic example of a function for which the Taylor series fails is:

#j(x) = { (0, " if " x = 0), (e^(-1/x^2), " if " x != 0) :}#

This has the property that the function and all of its derivatives exist and are

Inspired by this, consider the function:

#h(x) = { (0, " if " x = +-2), (e^(-x^2/(x^2-4)^2), " otherwise") :}#

Then

graph{e^(-x^2/(x^2-4)^2) [-5, 5, -2.5, 2.5]}

Given

Define

#g(x) = { (f(x), " if " x in (-oo, -2] uu [2, oo)), (f(x)+h(x)(x^9/256-f(x))(1+x^2/(x^2-4)^2+x^4/(2(x^2-4)^4)), " if " x in (-2, 2)) :}#

The idea here is that

graph{x^3-3x+e^(-x^2/(x^2-4)^2)*(x^9/256+3x-x^3)(1+x^2/(x^2-4)^2+x^4/(2(x^2-4)^4)) [-5, 5, -2.5, 2.5]}

The additional trick here is that the multiplier:

#(1+x^2/(x^2-4)^2+x^4/(2(x^2-4)^4))#

is a sufficiently good approximation to

As a result, we do not lose the smoothing properties of

It remains to show that the resulting function is monotonically increasing.