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Difference between revisions of "2007 Cyprus MO/Lyceum/Problem 12"

 
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==Problem==
 
==Problem==
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The function <math>f : \Re \rightarrow \Re</math> has the properties <math>f(0) = -1</math> and <math>f(xy)+f(x)+f(y)=x+y+xy+k \forall x,y \in \Re</math>, where <math>k \in \Re</math> is a constant. The value of <math>f(-1)</math> is
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The function <math>f : \mathbb{R} \rightarrow \mathbb{R}</math> has the properties <math>f(0) = -1</math> and <math>f(xy)+f(x)+f(y)=x+y+xy+k\ \ \ \forall x,y \in \Re</math>, where <math>k \in \Re</math> is a constant. The value of <math>f(-1)</math> is
  
 
<math> \mathrm{(A) \ } 1\qquad \mathrm{(B) \ } -1\qquad \mathrm{(C) \ } 0\qquad \mathrm{(D) \ } -2\qquad \mathrm{(E) \ } 3</math>
 
<math> \mathrm{(A) \ } 1\qquad \mathrm{(B) \ } -1\qquad \mathrm{(C) \ } 0\qquad \mathrm{(D) \ } -2\qquad \mathrm{(E) \ } 3</math>
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First, to determine the value of <math>k</math>, let <math>x=y=0</math>.
 
First, to determine the value of <math>k</math>, let <math>x=y=0</math>.
  
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<math>f(0\cdot0)+f(0)+f(0)=0+0+0\cdot0+k</math>
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<math>f(0\cdot0)+f(0)+f(0)=0+0+0\cdot0+k</math>, so <math>\displaystyle k = (-1)+(-1)+(-1) = - 3</math>.
  
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<math>(-1)+(-1)+(-1)=k</math>
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Now, to determine the value of <math>\displaystyle f(-1)</math>, let <math>x=-1</math> and <math>y=0</math>.
  
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<math>k=-3</math>
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<math>\displaystyle f(-1\cdot0)+f(-1)+f(0)=-1+0+0\cdot0-3</math>
  
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Now, to determine the value of <math>f(-1)</math>, let <math>x=-1</math> and <math>y=0</math>.
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<math>\displaystyle (-1)+f(-1)+(-1)=-4</math>
  
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<math>f(-1\cdot0)+f(-1)+f(0)=-1+0+0\cdot0-3</math>
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<math>\displaystyle f(-1)=-2\Longrightarrow\mathrm{ D}</math>
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<math>(-1)+f(-1)+(-1)=-4</math>
 
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<math>f(-1)=-2\Rightarrow\mathrm{ D}</math>
 
  
 
==See also==
 
==See also==
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*[[2007 Cyprus MO/Lyceum/Problems]]
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{{CYMO box|year=2007|l=Lyceum|num-b=11|num-a=13}}
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*[[2007 Cyprus MO/Lyceum/Problem 11|Previous Problem]]
 
  
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*[[2007 Cyprus MO/Lyceum/Problem 13|Next Problem]]
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[[Category:Introductory Algebra Problems]]

Latest revision as of 23:19, 18 January 2024

Problem

The function $f : \mathbb{R} \rightarrow \mathbb{R}$ has the properties $f(0) = -1$ and $f(xy)+f(x)+f(y)=x+y+xy+k\ \ \ \forall x,y \in \Re$, where $k \in \Re$ is a constant. The value of $f(-1)$ is

$\mathrm{(A) \ } 1\qquad \mathrm{(B) \ } -1\qquad \mathrm{(C) \ } 0\qquad \mathrm{(D) \ } -2\qquad \mathrm{(E) \ } 3$

Solution

First, to determine the value of $k$, let $x=y=0$.

$f(0\cdot0)+f(0)+f(0)=0+0+0\cdot0+k$, so $\displaystyle k = (-1)+(-1)+(-1) = - 3$.

Now, to determine the value of $\displaystyle f(-1)$, let $x=-1$ and $y=0$.

$\displaystyle f(-1\cdot0)+f(-1)+f(0)=-1+0+0\cdot0-3$

$\displaystyle (-1)+f(-1)+(-1)=-4$

$\displaystyle f(-1)=-2\Longrightarrow\mathrm{ D}$

See also

2007 Cyprus MO, Lyceum (Problems)
Preceded by
Problem 11 		Followed by
Problem 13
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
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