The only real solution to the equation $(x^2 + 100)^2 = (x^2 - 100)^2$ have how many digits in base 10 representation?
Step-by-Step Solution
Key Concept: Strict monotonicity allows replacing $f(f(x)) = x$ with $f(x) = x$ by composition properties.
Given $f(x) = \sqrt{x^2+100}$ is an increasing function, the equation $f(f(x)) = x$ can be written as $f(x) = x$, so $(f(x))^2 = x^2$. This gives $x^2 + 100 = x^3$, or $x^3 - x^2 - 100 = 0$, which factors as $(x-5)(x^2+4x+20) = 0$. Hence $x = 5$.
Correct Answer: 1