document.write( "Question 858186: How do you find the domain and range of the rational function: 5x^2+5/x^2+6x+9. Already found the asymptotes and x and y intercepts.\r
\n" ); document.write( "\n" ); document.write( "Vertical Asymptotes: x=-3
\n" ); document.write( "Horizontal Asymptotes: y=5
\n" ); document.write( "y-int.: (0,5/9)
\n" ); document.write( "x-int.: Does Not Exist\r
\n" ); document.write( "\n" ); document.write( "Use a graphing device to confirm your answer. (Use interval notation). \n" ); document.write( "

Algebra.Com's Answer #517105 by KMST(5328) $\"\"$ $\"About$

You can put this solution on YOUR website!
domain: all the real numbers except $\"-3\"$
\n" ); document.write( "range: [ $\"1%2F2\"$ , $\"infinity\"$ ) but I do not know how you would know.
\n" ); document.write( "Are you in a calculus class?
\n" ); document.write( "Are you studying derivatives?
\n" ); document.write( "
\n" ); document.write( "We know that $\"y=5%28x%5E2%2B1%29%2F%28x%2B3%29%5E2\"$ is always positive,
\n" ); document.write( "and that its value approaches $\"5\"$ at both ends,
\n" ); document.write( "but how low can it go?
\n" ); document.write( "Maybe you are supposed to calculate the derivative and find the minimum.
\n" ); document.write( "Maybe you are supposed to figure out the minimum some other way.
\n" ); document.write( "Apparently you are expected to use some graphing software to confirm.
\n" ); document.write( "
\n" ); document.write( "WITHOUT DERIVATIVES:
\n" ); document.write( "If $\"y=%285x%5E2%2B5%29%2F%28x%5E2%2B6x%2B9%29\"$ has a minimum $\"K\"$ ,
\n" ); document.write( " $\"%285x%5E2%2B5%29%2F%28x%5E2%2B6x%2B9%29-K=0\"$ has a single solution.
\n" ); document.write( " $\"%285x%5E2%2B5%29%2F%28x%5E2%2B6x%2B9%29-K=0\"$
\n" ); document.write( " $\"%285x%5E2%2B5-K%28x%5E2%2B6x%2B9%29%29%2F%28x%5E2%2B6x%2B9%29=0\"$
\n" ); document.write( "so $\"5x%5E2%2B5-K%28x%5E2%2B6x%2B9%29=0\"$ <--> $\"%285-K%29x%5E2-6Kx%2B%285-9K%29=0\"$ has a single solution.
\n" ); document.write( "That means the discriminant of that quadratic equation is zero.
\n" ); document.write( " $\"%286K%29%5E2-4%285-K%29%285-9K%29=0\"$
\n" ); document.write( " $\"36K%5E2-4%2825-50K%2B9K%5E2%29=0\"$
\n" ); document.write( " $\"36K%5E2-100%2B200K-36K%5E2=0\"$
\n" ); document.write( " $\"200K-100=0\"$
\n" ); document.write( " $\"200K=100\"$
\n" ); document.write( " $\"K=100%2F200\"$
\n" ); document.write( " $\"K=1%2F2\"$ The minimum of $\"y\"$ is $\"1%2F2\"$ .
\n" ); document.write( "
\n" ); document.write( "WITH DERIVATIVE:
\n" ); document.write( " $\"dy%2Fdx=10%283x-1%29%2F%28x%2B3%29%5E3\"$
\n" ); document.write( "and find out that the derivative is
\n" ); document.write( "undefined for $\"x=-3\"$ ,
\n" ); document.write( "negative for $\"-3%3Cx%3C1%2F3\"$ ,
\n" ); document.write( "positive for $\"x%3C-3\"$ and $\"x%3E1%2F3\"$ ,
\n" ); document.write( "and zero for $\"x=1%2F3\"$ ,
\n" ); document.write( "indicating a minimum of the function for $\"x=1%2F3\"$ .
\n" ); document.write( "Then, the minimum value for the function is

\n" ); document.write( "From that I know that for very negative values $\"y\"$ is barely above $\"5\"$ .
\n" ); document.write( "As $\"x\"$ increases,
\n" ); document.write( " $\"y\"$ grows without bounds as $\"x\"$ approaches $\"-3\"$ .
\n" ); document.write( "After that,
\n" ); document.write( " $\"y\"$ decreases to its $\"1%2F2=0.5\"$ minimum for $\"x=1%2F3\"$ ,
\n" ); document.write( "and then $\"y\"$ increases again towards $\"5\"$ .
\n" ); document.write( "Seeing all that in a graph requires viewing a bird's eye view and a zoomed-in view:
\n" ); document.write( " $\"graph%28600%2C300%2C-10%2C10%2C-5%2C45%2C5%28x%5E2%2B1%29%2F%28x%2B3%29%5E2%2C5%29\"$ $\"graph%28300%2C300%2C-100%2C100%2C-5%2C45%2C5%28x%5E2%2B1%29%2F%28x%2B3%29%5E2%2C5%29\"$ \n" ); document.write( "

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