<s>[INST] Inverse of fourth root function I have a function in the form
$$y=a\sqrt[^4]{b^2-(x-c)^2}, \qquad a, b, c\in \mathbb{R}$$
but I'm having trouble finding its inverse. That is, solving for $x$. The solution should seem pretty trivial with putting both sides to the $4^{\text{th}}$ power, rearranging, and applying the quadratic formula, but I can't seem to obtain the correct solution. Any suggestions? Thanks!
 [/INST] The equation is easier to work into the quadratic equation if we first get rid of radicals and avoid trying to "simplify" by using the $y^4/a^4$ rational. Expansion gives us the best view of a,b,c.
$$y=a\sqrt[^4]{b^2-(x-c)^2}\implies y^4=a^4\big(b^2-(c^2 - 2 c x + x^2)^2\big)\\
\implies 0=a^4 b^2 - a^4 c^2 + 2 a^4 c x - a^4 x^2-y^4\\
\implies      (a^4)x^2 - (2 a^4 c)x + (y^4 - a^4 b^2 + a^4 c^2) = 0\\
\implies x=  \frac{(2 a^4 c) \pm\sqrt{(2 a^4 c)^2-4(a^4)(y^4 - a^4 b^2 + a^4 c^2)}}{2a^4}\\
= \frac{(2 a^4 c) \pm\sqrt{4 a^8 b^2 - 4 a^4 y^4}}{2a^4} \\
\implies x = c \pm \frac{\sqrt{(a^2 b - y^2) (a^2 b + y^2)}}{a^2 }
\quad a,b,c,y\in\mathbb{R}\land a\ne 0\\
 x\in\mathbb{R}\iff y\le \sqrt{a^2b}$$
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