1972 AMC 12 Problem 29

Attempt Problem 29 of the 1972 AMC 12 below, then check your answer against the professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 1972 AMC 12 solutions, or check the answer key.

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29.

If

f(x)=\log\left(\dfrac{1+x}{1-x}\right)

for

-1\lt x\lt1,

then

f\left(\dfrac{3x+x^3}{1+3x^2}\right)

in terms of

f(x)

is:

-f(x)

2f(x)

3f(x)

[f(x)]^2

[f(x)]^3-f(x)

Answer: C

Concepts:logarithm function factoring substitution

Difficulty rating: 1860

Small hint:

Substitute the rational expression into

\frac{1+u}{1-u}

Big hint:

Its numerator and denominator factor as

(1+x)^3

and

(1-x)^3

Solution:

Let

u=\frac{3x+x^3}{1+3x^2}.

Then

\begin{aligned} \frac{1+u}{1-u} &=\frac{1+3x^2+3x+x^3} {1+3x^2-3x-x^3}\\ &=\left(\frac{1+x}{1-x}\right)^3. \end{aligned}

Consequently

f(u)=3\log\left(\frac{1+x}{1-x}\right)=3f(x).

Therefore, the correct answer is C.