If g(x)=1-x^2 and f(g(x))=(1-x^2)/(x^2) when x≠0, then f((1)/(2)) equals

1974 AMC 12 Problem 12

Attempt Problem 12 of the 1974 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 1974 AMC 12 solutions, or check the answer key.

All problems are used with official legal permission of the Mathematical Association of America (MAA).

12.

g(x)=1-x^2

and

f(g(x))=\frac{1-x^2}{x^2}

when

x\ne0,

then

f(\frac{1}{2})

equals

\frac{3}{4}

1

3

\frac{\sqrt2}{2}

\sqrt2

Choose a nonzero

x

for which

g(x)=\frac{1}{2}

The equation

1-x^2=\frac{1}{2}

determines

x^2

directly

g(x)=\frac{1}{2},

then

x^2=\frac{1}{2},

and either corresponding

x

is nonzero. Hence

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

Therefore, the correct answer is B.