What is the general solution to the equations f(g(x))=h(x) and g(f(x))=h(x), or how do you find the particular soltuion to the equations given a function f (given any f what is the general form of h(x) such that g(x) exists)?(adsbygoogle = window.adsbygoogle || []).push({});

My thoughts on the topic:

Specific example:

Suppose we have f(x)=x+1 and h(x)=x. Then g(x)=x-1.

Suppose instead h(x)=x+2. Then,

f(g(x))=g(x)+1=x+2

g(x)=x+1

g(f(x))=x+1+1=x+2 so it checks.

Now suppose h(x)=3x

f(g(x))=g(x)+1=3x

g(x)=3x-1

g(f(x))=3(x+1)-1=3x+2 doesn't check.

More general case:

Suppose now f(x)=x^{2}

f(g(x))=[g(x)]^{2}=h(x)

g(x)=sqrt(h(x))

g(f(x))=[squ]h(x^{2})=h(x)

So g(x) exists for a given h(x) if

h(x^{2})=[h(x)]^{2}

I don't think I can solve this any further. I feel resigned to the fact that I must check to see that this holds for some specific h(x) rather than finding the general solution.

Certainly, h(x)=x^{k}works. Any further thoughts?

Edit:

Oh by the way, h(x)=x of course works for all f provided that f has an inverse. So any general form should be able to reduce to h(x)=x.

**Physics Forums | Science Articles, Homework Help, Discussion**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Inverse Functions

**Physics Forums | Science Articles, Homework Help, Discussion**