Finding $g(x)$ , given $h(x)$ and $f(x)$

Question

find $g(x)$

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

$f(x) = x^{2}-6x+2$

$h(x) = 9x^{2}-24x+9$

My work:

$g(x)^{2}-6(g(x))+2 = h(x)$

$g(x)^{2}-6(g(x))-(9x^{2}-24x+7)$

And then what's next step?please provide explaination as well Thank You!!

3+- sqr(9-(-9x^2-24x+7))

3+- sqr(9+9x^2+24x-7)

Then what do i do?

Answer 1

Note $$\eqalign{f(x)&=(x-3)^2-7\\h(x)&=(3x-4)^2-7}$$ It follows that $$f(3x-1)=h(x).$$

Hope this helps.

Answer 2

Hint:

Let $y=g(x)$,

$$y^2-6y+2 = 9x^2-24x+9$$

$$y^2-6y-9x^2+24x-7=0$$ This is a quadratic equation in $y$, solve for $y$

$$y=\frac{6\pm \sqrt{6^2-4(-9x^2+24x-7)}}{2}$$