1967 AHSME Problems/Problem 14

Problem

Let $f(t)=\frac{t}{1-t}$ , $t \not= 1$ . If $y=f(x)$ , then $x$ can be expressed as

$\textbf{(A)}\ f\left(\frac{1}{y}\right)\qquad \textbf{(B)}\ -f(y)\qquad \textbf{(C)}\ -f(-y)\qquad \textbf{(D)}\ f(-y)\qquad \textbf{(E)}\ f(y)$

Solution

Since we know that $y=f(x)$ , we can solve for $y$ in terms of $x$ . This gives us

$y=\frac{x}{1-x}$

$\Rightarrow y(1-x)=x$

$\Rightarrow y-yx=x$

$\Rightarrow y=yx+x$

$\Rightarrow y=x(y+1)$

$\Rightarrow x=\frac{y}{y+1}$

Therefore, we want to find the function with $y$ that outputs $\frac{y}{y+1}$ Listing out the possible outputs from each of the given functions we get $f\left(\frac{1}{y}\right)=\frac{1}{y-1}$

$f(y)=\frac{1}{1-y}$

$f(-y)=\frac{-y}{y+1}$

$-f(y)=\frac{1}{y-1}$

$-f(-y)=\frac{y}{y+1}$

Since $-f(-y)=\frac{y}{y+1}=x$ the answer must be $\boxed{C}$ .

1967 AHSC (Problems • Answer Key • Resources)
Preceded by Problem 13	Followed by Problem 15
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1967 AHSME Problems/Problem 14

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