1981 AMC 12 Problem 17

Question

The function

f

is not defined for

x=0,

but, for all nonzero real numbers

x,

f(x)+2f\left(\frac1x\right)=3x.

The equation

f(x)=f(-x)

is satisfied by

exactly one real number

exactly two real numbers

no real numbers

infinitely many, but not all, nonzero real numbers

all nonzero real numbers

Apply the given relation again after replacing

x

by

\frac{1}{x}

Solve the two equations for

f(x)

, then compare its values at

x

and

-x

Replacing

x

by

\frac{1}{x}

gives

f(\frac{1}{x})+2f(x)=\frac{3}{x}.

Solving the two linear equations yields

f(x)=\frac2x-x.

Hence

f(-x)=-f(x),

so equality requires

f(x)=0.

Thus

\frac{2}{x}-x=0,

or

x^2=2.

Both

x=\sqrt2

and

x=-\sqrt2

work, giving exactly two real numbers.

Therefore, the correct answer is B.

Problem 17 in other years