If c is a constant and the function f defined by f(x)=(cx)/(2x+3), x≠-32, satisfies f(f(x))=x for all real numbers x except -32, then c is

1980 AMC 12 Problem 14

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14.

c

is a constant and the function

f

defined by

\begin{gathered} f(x)=\frac{cx}{2x+3},\\ x\ne-\frac32, \end{gathered}

satisfies

f(f(x))=x

for all real numbers

x

except

-\frac32,

then

c

-3

-\frac32

\frac32

3

not uniquely determined by the given information

Compute

f(f(x))

as a single rational expression

For the result to equal

x

identically, compare the coefficient of

x

in the denominator

Direct composition gives

f(f(x)) =\frac{c^2x}{(2c+6)x+9}.

For this to equal

x

identically, the denominator must be the constant

c^2.

Thus

2c+6=0

and

c^2=9,

both of which give

c=-3.

Therefore, the correct answer is A.