2006 AMC 12A Problem 18

Problem 18 of 25IntermediateAlgebra

The function

f

has the property that for each real number

x

in its domain,

\frac{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?

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Solution

Replacing

x

\frac{1}{x}

gives

f\!\left(\tfrac{1}{x}\right) + f(x) = \tfrac{1}{x}.

Together with

f(x) + f\!\left(\tfrac{1}{x}\right) = x,

this requires

x = \tfrac{1}{x},

x = \pm 1.

Both values are consistent, with

f(1) = \tfrac{1}{2}

and

f(-1) = -\tfrac{1}{2}.

So the largest possible domain is

\{-1, 1\}.

Thus, the correct answer is E.