# 2006 AMC 12A Problems/Problem 18

## Problem

The function $f$ has the property that for each real number $x$ in its domain, $1/x$ is also in its domain and $f(x)+f\left(\frac{1}{x}\right)=x$

What is the largest set of real numbers that can be in the domain of $f$? $\mathrm{(A) \ } \{x|x\ne 0\}\qquad \mathrm{(B) \ } \{x|x<0\}$ $\mathrm{(C) \ } \{x|x>0\}$ $\mathrm{(D) \ } \{x|x\ne -1\;\rm{and}\; x\ne 0\;\rm{and}\; x\ne 1\}$ $\mathrm{(E) \ } \{-1,1\}$

## Solution

Quickly verifying by plugging in values verifies that $-1$ and $1$ are in the domain. $f(x)+f\left(\frac{1}{x}\right)=x$

Plugging in $\frac{1}{x}$ into the function: $f\left(\frac{1}{x}\right)+f\left(\frac{1}{\frac{1}{x}}\right)=\frac{1}{x}$ $f\left(\frac{1}{x}\right)+ f(x)= \frac{1}{x}$

Since $f(x) + f\left(\frac{1}{x}\right)$ cannot have two values: $x = \frac{1}{x}$ $x^2 = 1$ $x=\pm 1$

Therefore, the largest set of real numbers that can be in the domain of $f$ is $\{-1,1\} \Rightarrow E$

## Solution 2

We know that $f(x) + f \left(\frac{1}{x}\right) = x.$ Plugging in $x = \frac{1}{x}$ we get $$f \left(\frac{1}{x}\right) + f \left(\frac{1}{\frac{1}{x}}\right) = \frac{1}{x}$$ $$f \left(\frac{1}{x}\right) + f(x) = \frac{1}{x}.$$

Also notice $$f \left(\frac{1}{x}\right) + f(x) = x$$ by the commutative property(this is the same as the equation given in the problem. We are just rearranging). So we can set $\frac{1}{x} = x$ which gives us $x = \pm 1$ which is answer option $\boxed{\mathrm{(E) \ } \{-1,1\}}.$

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