Difference between revisions of "1967 AHSME Problems/Problem 14"

Revision as of 13:16, 10 August 2020

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

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

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

$\Rightarrow y-yx=1$

$\Rightarrow yx=y-1$

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

Therefore, we want to find the function with $y$ that outputs $\frac{y-1}{y}$ 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 $\boxed{E}$ .

1967 AHSME (Problems • Answer Key • Resources)
Preceded by Problem 13	Followed by Problem 15
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 • 26 • 27 • 28 • 29 • 30
All AHSME Problems and Solutions

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.

@@ Line 21: / Line 21: @@
 <math>\Rightarrow x=\frac{y-1}{y}</math>
+Therefore, we want to find the function with <math>y</math> that outputs <math>\frac{y-1}{y}</math>
+Listing out the possible outputs from each of the given functions we get
+<math>f\left(\frac{1}{y}\right)=\frac{1}{y-1}</math>
+<math>f(y)=\frac{1}{1-y}</math>
+<math>f(-y)=\frac{-y}{y+1}</math>
+<math>-f(y)=\frac{1}{y-1}</math>
+<math>-f(-y)=\frac{y}{y+1}</math>
+Since <math>-f(-y)=\frac{y}{y+1}=x</math> the answer must be <math>\boxed{E}</math>.
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Difference between revisions of "1967 AHSME Problems/Problem 14"

Revision as of 13:16, 10 August 2020

Problem

Solution

See also