So Many Functions!

Algebra Level 3

Let there be functions $a(x), b(x),$ and $c(x)$ :

$a(x) = x² +4x +4$ , $b(x)= \frac{1}{x⁴ +4x² +4}$ , $c(x)=x²$

Then, $(a⁻¹○b○c⁻¹)(x)$ can be expressed as two different functions, $S₁(x)$ or $S₂(x)$ .

Then, let $d(x)= S₁(x)² + S₂(x)²$ . Find the sum of all complex zeroes of $d(x)$ .

Definitions:

$(f○g○h)(x)$ means $f(g(h(x)))$

$f⁻¹(x)$ means the inverse of the function $f(x)$ .

1 solution

Tom Engelsman
Aug 9, 2020

Let $a^{-1}(x) = \sqrt{x} -2$ and $c^{-1}(x) = \sqrt{x}$ . The three-fold composition computes to:

$a^{-1}(b(c^{-1}(x))) = \frac{1}{x+2} - 2 \Rightarrow S_{1}(x) = \frac{1}{x+2}, S_{2}(x) = -2.$

Taking $d(x) = S_{1}^{2}(x) + S_{2}^{2}(x) = \frac{1}{(x+2)^{2}} + 4 = 0$ , the complex zeros compute to:

$\frac{1}{x+2} = \pm 2i \Rightarrow x = -2 \pm \frac{1}{2} i$

and sum to $\boxed{-4}.$