2008 Mock ARML 1 Problems/Problem 1

Problem

Compute all real values of such that .

Solution

Let ; then . Because is increasing on , $f(f(x))=x\Longrightarrowf(x)=x$ (Error compiling LaTeX. Unknown error_msg). Using this we can show . Using your favorite method, solve for . However, since , and because the Square Root function's range does not include negative numbers, it follows that the negative root is extraneous, and thus we have . The other roots we can verify are not real.

See also

2008 Mock ARML 1 (Problems, Source)
Preceded by First question	Followed by Problem 2
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8

square both sides twice leaving:

{x+4}=(x-4)^2

then subtract x-4 to set to 0 (from x^2-8x^2+16)

using the rational roots theorem, we get the quadratics:

(x^2-x-4)(x^2+x-3)

Solve: -1+/-sqrt{13}/2 1+/-sqrt{17}/2

Seeing that negative roots are extraneous we have:

1+sqrt{17}/2 and -1+sqrt{13}/2 as the answers.