Vtyjkyuk

Suppose I have the function

$$f(x) = frac203frac1x - frac23$$

and I would like to mirror it at the line

$$l(x) = -frac53x+frac233$$

to get a function $g(x)$. Is it possible to get a closed form solution of such a $g$ in terms of $x$? As per this question I know how the mirror looks like. Now can I write this is as a closed form solution?

The picture I have in mind is the following:

where I have the lower part and the linear function. Now I would like to get the upper one.

Given a line $ax+by=c$, the reflection transformation is

$$beginalign
x &Rightarrow frac(b^2-a^2)x - 2aby + 2aca^2+b^2\
y &Rightarrow frac-2abx + (a^2-b^2)y + 2bca^2+b^2
endalign$$

We have $5x+3y=23$, so the reflection transformation here is

$$beginalign
x &Rightarrow frac-8x-15y+11517 \
y &Rightarrow frac-15x+8y+6917
endalign
$$

Before we apply this we should probably switch the hyperbola to standard form, by multiplying the whole thing by $3x$ and rearranging:

$$3xy + 2x = 20$$

We apply the transformation:

$$3frac-8x-15y+11517frac-15x+8y+6917 + 2frac-8x-15y+11517 = 20$$

And then simplify (multiply through by $17^2=289$, expand, combine):

$$360x^2+483xy-360y^2-7103x-855y=-21935$$

Then, we can find the "function" by solving for $y$, which we can do by grouping on $y$ exponents and then using the quadratic formula:

$$(-360)y^2 + (483x-855)y + (360x^2 -7103x + 21935) = 0$$

$$beginalign
y&=frac-(483x-855)pmsqrt(483x-855)^2-4(-360)(360x^2 -7103x + 21935)2(-360)\
&=frac161x-285pm17sqrt289x^2-4250x+12425240
endalign$$

This has two parts: notice the $pm$ in there, giving two valid $y$ values for most values of $x$.

The discriminant has roots $frac125pm40sqrt217$, which are the $x$ values of our branch points. As you can see in the graph below, even the small reflected part that would close the lens shape fails to be a single function.