Composite functions

veroL

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Jul 19, 2018
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Hello.

I need to find the possible values of a and b when (f°g)(x)=(g°f)(x) if f(x)=x^2 and g(x)=ax+b.
I'm able to find the composite functions ( (f°g)(x)=(ax+b)^2 and (g°f)(x)=a(x^2)+b but I'm trully stumped on how to solve the equation...

Thanks in advance!
 
Hello.

I need to find the possible values of a and b when (f°g)(x)=(g°f)(x) if f(x)=x^2 and g(x)=ax+b.
I'm able to find the composite functions ( (f°g)(x)=(ax+b)^2 and (g°f)(x)=a(x^2)+b but I'm trully stumped on how to solve the equation...

Thanks in advance!
That's very good.

Have you considered pondering \(\displaystyle (ax+b)^{2} = ax^{2}+b\)? What's the rule for two polynomials being equivalent?
 
Two polynomials are considered equivalent, if they each have the same roots. There are two ways this can happen.


1) All corresponding coefficients are equal:

2x^2 + 4x + 8

4(x + 2) + 2x^2


2) Each coefficient in one polynomial is a constant multiple of the corresponding coefficient in the other polynomial:

3x^2 + 5x - 2

6x^2 + 10x - 4


Here's a brief lesson, with a worked example showing the first case above. :cool:
 
That's very good.

Have you considered pondering \(\displaystyle (ax+b)^{2} = ax^{2}+b\)? What's the rule for two polynomials being equivalent?
Based on mmm4444Bot's post I need to ask what you mean by equivalent? Personally I would have said equal.
 
Two polynomials are considered equivalent, if they each have the same roots. There are two ways this can happen.


1) All corresponding coefficients are equal:

2x^2 + 4x + 8

4(x + 2) + 2x^2


2) Each coefficient in one polynomial is a constant multiple of the corresponding coefficient in the other polynomial:

3x^2 + 5x - 2

6x^2 + 10x - 4


Here's a brief lesson, with a worked example showing the first case above. :cool:

I did not know that rule! XD

So in my case, that would mean \(\displaystyle a^{2}x^{2}+2axb+b^{2}=ax^{2}+b\)

So \(\displaystyle (a)^{2}\) must be equal to a because they are both coefficients to \(\displaystyle (x)^{2}\)??

And then once I've found a (0 or 1), I just have to plug it in one of the equations to find b?
 
… that would mean \(\displaystyle a^{2}x^{2}+2axb+b^{2}=ax^{2}+b\)
You got it. Although, I would have written the first-degree term on the left as 2abx or (2ab)x, to more clearly show the coefficient expression.


So \(\displaystyle (a)^{2}\) must be equal to a because they are both coefficients to \(\displaystyle (x)^{2}\)??

And then once I've found a (0 or 1), I just have to plug it in one of the equations to find b?
Yes, a^2 must equal a.

But also, the first-degree terms must have equal coefficients, and the two contants must also be equal. I would consider all of this, before trying substitutions. :cool:
 
You got it. Although, I would have written the first-degree term on the left as 2abx or (2ab)x, to more clearly show the coefficient expression.


Yes, a^2 must equal a.

But also, the first-degree terms must have equal coefficients, and the two contants must also be equal. I would consider all of this, before trying substitutions. :cool:

Good point!

Thank you so much!! :D
 
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