Difference between revisions of "1997 AIME Problems/Problem 12"
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== Problem == | == Problem == | ||
− | The function <math>f</math> defined by <math>f(x)= \frac{ax+b}{cx+d}</math>, where <math>a</math>,<math>b</math>,<math>c</math> and <math>d</math> are nonzero real numbers, has the properties <math>f(19)=19</math>, <math>f(97)=97</math> and <math>f(f(x))=x</math> for all values except <math>\frac{-d}{c}</math>. Find the unique number that is not in the range of <math>f</math>. | + | The [[function]] <math>f</math> defined by <math>f(x)= \frac{ax+b}{cx+d}</math>, where <math>a</math>,<math>b</math>,<math>c</math> and <math>d</math> are nonzero real numbers, has the properties <math>f(19)=19</math>, <math>f(97)=97</math> and <math>f(f(x))=x</math> for all values except <math>\frac{-d}{c}</math>. Find the unique number that is not in the range of <math>f</math>. |
== Solution == | == Solution == | ||
− | {{ | + | First, we note that <math>e = \frac ac</math> is the horizontal [[asymptote]] of the function, and since this is a linear function over a linear function, the unique number not in the range of <math>f</math> will be <math>e</math>. <math>\frac{ax+b}{cx+d} = \frac{b-\frac{cd}{a}}{cx+d} + \frac{a}{c}</math>. [[Without loss of generality]], let <math>c=1</math>, so the function becomes <math>\frac{b- \frac{d}{a}}{x+d} + e</math>. |
+ | |||
+ | (Considering <math>\infty</math> as a limit) By the given, <math>f(f(\infty)) = \infty</math>. <math>\lim_{x \rightarrow \infty} f(x) = e</math>, so <math>f(e) = \infty</math>. <math>f(x) \rightarrow \infty</math> as <math>x</math> reaches the vertical [[asymptote]], which is at <math>-\frac{d}{c} = -d</math>. Hence <math>e = -d</math>. Substituting the givens, we get | ||
+ | |||
+ | <cmath>\begin{eqnarray*} | ||
+ | 17 &=& \frac{b - \frac da}{17 - e} + e\\ | ||
+ | 97 &=& \frac{b - \frac da}{97 - e} + e\\ | ||
+ | b - \frac da &=& (17 - e)^2 = (97 - e)^2\\ | ||
+ | 17 - e &=& \pm (97 - e) | ||
+ | \end{eqnarray*}</cmath> | ||
+ | |||
+ | Clearly we can discard the positive root, so <math>e = \box{58}</math>. | ||
== See also == | == See also == | ||
{{AIME box|year=1997|num-b=11|num-a=13}} | {{AIME box|year=1997|num-b=11|num-a=13}} | ||
+ | |||
+ | [[Category:Intermediate Algebra Problems]] |
Revision as of 21:30, 23 November 2007
Problem
The function defined by , where ,, and are nonzero real numbers, has the properties , and for all values except . Find the unique number that is not in the range of .
Solution
First, we note that is the horizontal asymptote of the function, and since this is a linear function over a linear function, the unique number not in the range of will be . . Without loss of generality, let , so the function becomes .
(Considering as a limit) By the given, . , so . as reaches the vertical asymptote, which is at . Hence . Substituting the givens, we get
Clearly we can discard the positive root, so $e = \box{58}$ (Error compiling LaTeX. ! Missing number, treated as zero.).
See also
1997 AIME (Problems • Answer Key • Resources) | ||
Preceded by Problem 11 |
Followed by Problem 13 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |