document.write( "Question 1209949: The function f : \mathbb{R} \rightarrow \mathbb{R} satisfies
\n" ); document.write( "f(x) f(y) - f(xy) = -2x - 6y + 10
\n" ); document.write( "for all x, y \in \mathbb{R}. Find f(x). \n" ); document.write( "

Algebra.Com's Answer #851218 by greenestamps(13200) $\"\"$ $\"About$

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\n" ); document.write( "There is no function that satisfies the given condition.

\n" ); document.write( "That is easily seen by observing that the expression on the left is symmetric in x and y but the expression on the right is not. In other words, switching the values of x and y on the left gives the same result but switching them on the right does not.

\n" ); document.write( "For example....

\n" ); document.write( "(x,y) = (1,0)

\n" ); document.write( " $\"f%281%29%2Af%280%29-f%280%29=-2%2B0%2B10\"$
\n" ); document.write( " $\"f%281%29%2Af%280%29=f%280%29%2B8\"$
\n" ); document.write( " $\"f%281%29=%28f%280%29%2B8%29%2Ff%280%29=1%2B8%2Ff%280%29\"$

\n" ); document.write( "(x,y) = (0,1)

\n" ); document.write( " $\"f%280%29%2Af%281%29-f%280%29=0-6%2B10\"$
\n" ); document.write( " $\"f%281%29%2Af%280%29=f%280%29%2B4\"$
\n" ); document.write( " $\"f%281%29=%28f%280%29%2B4%29%2Ff%280%29=1%2B4%2Ff%280%29\"$

\n" ); document.write( " $\"1%2B8%2Ff%280%29\"$ is never equal to $\"1%2B4%2Ff%280%29\"$

\n" ); document.write( " \n" ); document.write( "

\n" );