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The problem

$f(x)$ and $g(x)$ are defined over the real number set $\mathbb{R}$ as follows:

$$ \begin{split} g(x) &= 1-x+x^2\\ f(x) &= ax+b \end{split} $$

If $g(f(x)) = 9x^2 - 9x + 3$, determine all the possible values of $a$ and $b$.

Basically i'm completely thrown by this. How would you begin to work out the values of $a$ & $b$? What sort of tests?

I get functions, but have just never been asked something like this, any help simplifying this would be much appreciated.

I'm not looking for all the possible values of $a$ & $b$. Just someone to tell me what direction to go to find that out.

Many thanks

gt6989b
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Lee Brindley
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1 Answers1

5

HINT:

$g(x)= 1-x+x^2$ and $ f(x)=ax+b$

$\implies g(f(x))=g(ax+b)$ $=1-(ax+b)+(ax+b)^2=a^2x^2+x(2ab-a)+1-b+b^2$

Again, $g(f(x)) = 9x^2 - 9x + 3$

Equate the coefficients of the different powers of $x$

lab bhattacharjee
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