Let $f(x)=16x+3$. Find the sum of all $x$ that satisfy the equation $f^{-1}(x)=f((2x)^{-1})$.
Applying $f$ to both sides of the equation $f^{-1}(x) = f((2x)^{-1})$, we get $f(f^{-1}(x)) = f(f((2x)^{-1}))$.  By definition of the inverse function, $f(f^{-1}(x)) = x$, and \[f(f((2x)^{-1})) = f \left( f \left( \frac{1}{2x} \right) \right) = f \left( \frac{16}{2x} + 3 \right) = f \left( \frac{8}{x} + 3 \right) = f \left( \frac{3x + 8}{x} \right) = 16 \cdot \frac{3x + 8}{x} + 3 = \frac{51x + 128}{x}.\]Hence, \[x = \frac{51x + 128}{x}.\]Then $x^2 = 51x + 128$, or $x^2 - 51x - 128 = 0$.  Vieta's formula tells us that the sum of the roots of a quadratic $ax^2+bx+c$ is $-\frac{b}{a}$, so in this case, the sum of the roots is $\boxed{51}$.
51
Level 4
Intermediate Algebra
Problem: Let $f(x)=16x+3$. Find the sum of all $x$ that satisfy the equation $f^{-1}(x)=f((2x)^{-1})$.
Solution:<|START_OF_LATENT|><|PRIOR_PREFIX|>To solve the equation $f^{-1}(x)=f((2x)^{-1})$, we first need to find the inverse function $f^{-1}(x)$. Given $f(x)=16x+3$, we set $y=f(x)$, which gives us $y=16x+3$. Rearranging this equation to solve for $x$ yields $x=\frac{y-3}{16}$. Thus, $f^{-1}(x)=\frac{x-3}{16}$.

Next, we compute $f((2x)^{-1})$. We first find $(2x)^{-1}=\frac{1}{2x}$. Substituting this into the function $f$, we have $f\left(\frac{1}{2x}\right)=16\left(\frac{1}{2x}\right)+3=\frac{16}{2x}+3=\frac{8}{x}+3$.

Now we set the two expressions equal to each other: 
$$\frac{x-3}{16}=\frac{8}{x}+3.$$

To eliminate the fractions, we multiply through by $16x$:
$$x(x-3)=16(8+3x).$$

Expanding both sides gives:
$$x^2 - 3x = 128 + 48x.$$

Rearranging this equation leads to:
$$x^2 - 3x - 48x - 128 = 0,$$
which simplifies to:
$$x^2 - 51x - 128 = 0.$$

Next, we apply the quadratic formula $x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$, where $a=1$, $b=-51$, and $c=-128$. Calculating the discriminant:
$$b^2 - 4ac = (-51)^2 - 4(1)(-128) = 2601 + 512 = 3113.$$

Now we find the roots:
$$x=\frac{51\pm\sqrt{3113}}{2}.$$

The sum of the roots of the quadratic equation can be directly obtained from the formula $-\frac{b}{a}$, which gives us $51$. Thus, the final answer is $51$.<|END_OF_LATENT|> The final answer is $51$.