1975 AMC 12 Problem 21

Question

Suppose

f(x)

is defined for all real numbers

x;

f(x)\gt0

for all

x;

and

f(a)f(b)=f(a+b)

for all

a

and

b.

Which of the following statements are true?

I.

f(0)=1

II.

f(-a)=\frac{1}{f(a)}

for all

a

III.

f(a)=\sqrt[3]{f(3a)}

for all

a

IV.

f(b)\gt f(a)

if

b\gt a

\mathrm{III}

and

\mathrm{IV}

only

\mathrm{I},

\mathrm{III}

and

\mathrm{IV}

only

\mathrm{I},

\mathrm{II}

and

\mathrm{IV}

only

\mathrm{I},

\mathrm{II}

and

\mathrm{III}

only

All are true.

Substitute

a=0,\ b=-a,

and then add

a

three times

Test the monotonicity claim with the constant function

f(x)=1

Taking

a=0

and using positivity gives

f(0)=1.

Taking

b=-a

gives

f(a)f(-a)=1.

Also

f(3a)=f(a)^3,

so positivity permits the positive cube root in III. But

f(x)=1

satisfies the functional equation and is not strictly increasing, so IV need not hold.

Therefore, the correct answer is D.

Problem 21 in other years