1953 AMC 12 Problem 35

Attempt Problem 35 of the 1953 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 1953 AMC 12 solutions, or check the answer key.

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35.

If

f(x)=\dfrac{x(x-1)}2,

then

f(x+2)

equals:

f(x)+f(2)

(x+2)f(x)

x(x+2)f(x)

\dfrac{xf(x)}{x+2}

\dfrac{(x+2)f(x+1)}x

Answer: E

Concepts:function notation algebraic identity

Difficulty rating: 1450

Small hint:

Write explicit formulas for both

f(x+2)

and

f(x+1)

Big hint:

The factor

x

in

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

cancels in one choice

Solution:

Directly,

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

Also

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

so

\begin{gathered} \frac{(x+2)f(x+1)}x\\ =\frac{(x+2)(x+1)}2. \end{gathered}

Thus, the correct answer is E.