document.write( "Question 547530: determine the equations of any vertical asymptotes and the values of x for any holes in the graph of each rational function.\r
\n" ); document.write( "\n" ); document.write( "f(x)=2x^2-x-10/(2x-5) \n" ); document.write( "

Algebra.Com's Answer #356437 by KMST(5328) $\"\"$ $\"About$

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$\"f%28x%29=%282x%5E2-x-10%29%2F%282x-5%29\"$ is a rational function (meaning a function involving at most polynomials and maybe quotients of polynomials). If you thought you would not have to factor and divide polynomials ever again, you were wrong.
\n" ); document.write( "With rational functions, you have to factor, or divide often. It is essential figure out what happens when a denominator is zero. For values of x that make a denominator zero, the function is undefined. It could be a hole, or it could be a vertical asymptote.
\n" ); document.write( "Factoring, we find:
\n" ); document.write( " $\"f%28x%29=%282x%5E2-x-10%29%2F%282x-5%29=%282x-5%29%28x%2B2%29%2F%282x-5%29\"$
\n" ); document.write( " $\"2x-5=0\"$ <---> $\"x=5%2F2\"$ and the function does not exist.
\n" ); document.write( "For any other x, $\"f%28x%29=x%2B2\"$
\n" ); document.write( "So $\"x=5%2F2\"$ is a hole in the line $\"y=x%2B2\"$ .
\n" ); document.write( "No vertical asymptote. Just a hole. \n" ); document.write( "

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