Using function compositions to determine variable values

Question

Let $f,g:$ $R \rightarrow R$ where $g(x)$ $=$ $1$ $-$ $x$ $+$ $x^2$ and $f(x)$ $=$ $ax$ $+$ $b$

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

So far, I have "fit" $f(x)$ into $g(x)$ as follows: $g(f(x))$ $=$ $(ax + b)^2$ $-$ $(ax+b)$ $+$ $1$

Is there a relationship between compositions and inverse functions I should be using to isolate $a$ and $b$? How does one extract those variables in order to determine their value?

Thank you!

Answer 1

First expand the expression and regroup the components.

$$g(f(x)) = 1 − (ax+b) + (ax+b)^2$$

$$g(f(x)) = 1 − ax − b + a^2x^2 + 2abx + b^2$$

$$g(f(x)) = (1 − b + b^2) + a(2b − 1)x + a^2x^2$$

We know that

$$g(f(x)) = 3 − 9x + 9x^2$$

Therefore

$$(1 − b + b^2) + a(2b − 1)x + a^2x^2 = 3 − 9x + 9x^2$$

The system to solve is

$$a^2=9$$

$$a(2b − 1)=−9$$

$$1 − b + b^2=3$$

So $a = \pm 3$, then

• if $a = 3$, then $b = -1$, and the last equation does not satisfy
• if $a = -3$, then $b = 2$, and the last equation does satisfy.

Finally, the solution is $a = -3$, $b = 2$.