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# Composite Functions

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1. If f(x)=ax+b and f^(3)(x)=64x+21 find the values of a and b. Suggest a rule for f^(n)(x)

Would f^3(x) be a^3x+3b and f^n(x)= 4^n(x)+7n?

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2. (Original post by Mr Pussyfoot)
If f(x)=ax+b and f^(3)(x)=64x+21 find the values of a and b. Suggest a rule for f^(n)(x)

Would f^3(x) be a^3x+3b and f^n(x)= 4^n(x)+7n?

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It certainly fits so i believe it's correct
3. (Original post by Mr Pussyfoot)
If f(x)=ax+b and f^(3)(x)=64x+21 find the values of a and b. Suggest a rule for f^(n)(x)

Would f^3(x) be a^3x+3b and f^n(x)= 4^n(x)+7n?

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No.

f(x) = ax+b

So,

and

And take it from there.
4. It may be useful to prove your suggested rule by induction, quite easy.
5. (Original post by ghostwalker)
No.

f(x) = ax+b

So,

and

And take it from there.
Would the rule be a^(n)x+b(a^(n-1)+a^(n-2)...a^0)?

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6. (Original post by Mr Pussyfoot)
Would the rule be a^(n)x+b(a^(n-1)+a^(n-2)...a^0)?
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Looks good. Notice that the constant term is imply "b" times a G.P., so can be simplified.
7. (Original post by ghostwalker)
Looks good. Notice that the constant term is imply "b" times a G.P., so can be simplified.
I see, so the fully simplified version would come down to a^nx+b(1-a^n/1-a) I'm guessing

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8. And on another note, if a f(x)=1/(1-x), would f^2(x)= 1/(2+x)? Nevermind, this is incorrect lol

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9. (Original post by Mr Pussyfoot)
And on another note, if a f(x)=1/(1-x), would f^2(x)= 1/(2+x)? Nevermind, this is incorrect lol

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No, it wouldn't be. It would be

10. I've gotten this far on question 15 but have no idea what to do next, should I substitute the values of x into f^-1(x)=f^2(x)?

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11. (Original post by Zacken)
No, it wouldn't be. It would be
Yeah I realised this after a while

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12. (Original post by Mr Pussyfoot)
I've gotten this far on question 15 but have no idea what to do next, should I substitute the values of x into f^-1(x)=f^2(x)?
Nope, nope, nope. This is why I always encourage writing
13. (Original post by Mr Pussyfoot)
I see, so the fully simplified version would come down to a^nx+b(1-a^n/1-a) I'm guessing

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That's correct.

Though you can of course simplify further by putting in the actual values of a and b.

Then as a check set n=3, and see if if gives the desired result. You can also check n=1, to see it gives ax+b, for whatever your values of a,b are.
14. (Original post by Zacken)
Nope, nope, nope. This is why I always encourage writing

Okay I arrived at this quadratic, should I follow on to solve for x? I notice that the coefficient of x^2 is similar to what the question is asking for, is there a link?

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Attached Images

15. (Original post by Mr Pussyfoot)

Okay I arrived at this quadratic, should I follow on to solve for x? I notice that the coefficient of x^2 is similar to what the question is asking for, is there a link?

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You want the quadratic to be satisfied for all values of , that is, you need all of the coefficients to be equal to 0 so that the whole thing is equal to 0, i.e: solving for will give two values of in terms of and such that the quadratic holds, you want the quadratic to be identically 0 for all (think in terms of: you want it to be an identity not an equation), so anyway.

So you need all of .

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