What is the function g(x) given h=g \circ f and f(x)=x+1, h(x)=x^3+3x+1?

[Дьявол]

Homework Statement

Find the function g(x), if there are $h=g \circ f$

f(x)=x+1, $x \in \mathbb{R}$ and $h(x)=x^3+3x+1$

Homework Equations

$$(g \circ f)(x)=g(f(x))=h(x)$$

$$f \circ g \neq g \circ f$$

The Attempt at a Solution

$$h=g \circ f$$

$$h(x)=g(f(x))$$

$$x^3+3x+1=g(x+1)$$

Is there any way that I will directly find the result of g(x), or I should guess and try some things? I tried something at home, but useless. Please help me!

 

[Office_Shredder]

My guess is g is going to be a polynomial. You can use the degree of f and h to find what the degree of g is going to be, then plug f into a generic polynomial of that degree and see what conditions the coefficients have to satisfy

 

[Dick]

Well, you can figure g ought to be a polynomial function of degree 3, right? So put g(x+1)=A(x+1)^3+B(x+1)^2+C(x+1)+D. If you set that equal to x^3+3x+1, can you find A,B,C and D? You should get four equations in four unknowns if you equate the powers of x. Of course, you can also be clever and pick some special values of x that might make the job easier (like x=-1).

 

[Дьявол]

Thanks for the posts.

I found A=1, B=-3, C=6, D=-3.
$$g(x+1)=(x+1)^3-3(x+1)^2+6(x+1)-3$$
So
$$g(x)=x^3-3x^2+6x-3$$

Yes it is correct.

How will x=-1 make the job easier?

 

[Dick]

Thanks for the posts.

I found A=1, B=-3, C=6, D=-3.
$$g(x+1)=(x+1)^3-3(x+1)^2+6(x+1)-3$$
So
$$g(x)=x^3-3x^2+6x-3$$

Yes it is correct.

How will x=-1 make the job easier?

If you put x=-1, you get D=-3 directly. You can also see A must be 1 without doing any any algebra. So you really only need to solve for B and C.

 

[gabbagabbahey]

$$x^3+3x+1=g(x+1)$$

Is there any way that I will directly find the result of g(x), or I should guess and try some things? I tried something at home, but useless. Please help me!

An alternative method would be to just make the substitution u=x+1 (and hence x=u-1) into this equation and find g(u), then rename u to x and your done...You wouldn't even need to expand the polynomial, you could just leave it as a polynomial in powers of (x-1)

 

[Дьявол]

Thanks for the replys.
gabbagabbahey, do you think like $(u-1)^3+3(u-1)+1=g(u)$ ?

I got the same result $u^3-3u^2+6u-3=g(u)$, so if I write $g(x)=u^3-3u^2+6u-3$. Thanks again.

 

[gabbagabbahey]

Yes, $(u-1)^3+3(u-1)+1=g(u)$ so automatically, $g(x)=(x-1)^3+3(x-1)+1$ and you could leave the answer just like that. (or you could expand it in powers of x and get $g(x)=x^3-3x^2+6x-3$)

 

[Dick]

An alternative method would be to just make the substitution u=x+1 (and hence x=u-1) into this equation and find g(u), then rename u to x and your done...You wouldn't even need to expand the polynomial, you could just leave it as a polynomial in powers of (x-1)

That's even more direct. Good observation.