document.write( "Question 1019976: please help solve this question.
\n" ); document.write( " Consider the function
\n" ); document.write( "f(x)=(x^2-1)/(x-3)
\n" ); document.write( " a. Find the domain for the function in interval notations
\n" ); document.write( " b. Find all vertical, horizontal and slant asymptotes
\n" ); document.write( " c. Sketch the function
\n" ); document.write( " d. Solve the inequality
\n" ); document.write( " (x^2-1)/(x-3)<1
\n" ); document.write( " \n" ); document.write( "

Algebra.Com's Answer #635929 by KMST(5328) $\"\"$ $\"About$

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$\"f%28x%29=%28x%5E2-1%29%2F%28x-3%29\"$
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\n" ); document.write( "a. Since the expression for $\"%28x%5E2-1%29%2F%28x-3%29\"$ is defined for all real values of $\"x\"$ such that $\"x%3C%3E3\"$ ,
\n" ); document.write( "the domain for the function in interval notation is
\n" ); document.write( " $\"%22%28%22\"$ $\"-infinity\"$ $\"%22%2C+3+%29+U+%28+3+%2C%22\"$ $\"infinity\"$ $\"%22%29%22\"$ .
\n" ); document.write( "
\n" ); document.write( "b. Find all vertical, horizontal and slant asymptotes
\n" ); document.write( " $\"x=3\"$ is a vertical asymptote,
\n" ); document.write( "because the function does not exist for $\"x=3\"$ ,
\n" ); document.write( "but

.
\n" ); document.write( "

,
\n" ); document.write( "so $\"highlight%28y=x%2B3%29\"$ is the slant asymptote, because
\n" ); document.write( " $\"lim%28x-%3Einfinity%2Cf%28x%29-%28x%2B3%29%29=lim%28x-%3Einfinity%2C8%2F%28x-3%29%29=0\"$ .
\n" ); document.write( "That means the graph function \"hugs\" the slanted line representing $\"y=x%2B3\"$ towards the left and right ends of the graph, and off course, there is no horizontal asymptote.
\n" ); document.write( "We even know that $\"f%28x%29-%28x%2B3%29=8%2F%28x-3%29%3E0\"$ for $\"x%3E3\"$ , meaning the curve is above the asymptote,
\n" ); document.write( "and $\"f%28x%29-%28x%2B3%29=8%2F%28x-3%29%3C0\"$ for $\"x%3C3\"$ , meaning the curve is below the asymptote.
\n" ); document.write( "
\n" ); document.write( "c. Sketch the function
\n" ); document.write( "We know the asymptotes.
\n" ); document.write( "To sketch it, we may want to know
\n" ); document.write( "1)where the function is negative, zero, and positive, and
\n" ); document.write( "2) maxima and minima.
\n" ); document.write( "Since $\"f%28x%29=%28x%5E2-1%29%2F%28x-3%29=%28x%2B1%29%28x-1%29%2F%28x-3%29\"$ ,
\n" ); document.write( " $\"x=-1\"$ and $\"x=1%29%29%29+make+the+numerator%2C+and+%7B%7B%7Bf%28x%29\"$ zero.
\n" ); document.write( "For $\"x%3C-1\"$ , $\"system%28x%2B1%3C0%2Cx-1%3C0%2Cx-3%3C0%29\"$ ---> $\"f%28x%29%3C0\"$ .
\n" ); document.write( "For $\"-1%3Cx%3C1\"$ , $\"system%28x%2B1%3E0%2Cx-1%3C0%2Cx-3%3C0%29\"$ ---> $\"f%28x%29%3E0\"$ .
\n" ); document.write( "For $\"1%3Cx%3C3\"$ , $\"system%28x%2B1%3E0%2Cx-1%3E0%2Cx-3%3C0%29\"$ ---> $\"f%28x%29%3C0\"$ .
\n" ); document.write( "For $\"x%3E3\"$ , $\"system%28x%2B1%3E0%2Cx-1%3E0%2Cx-3%3E0%29\"$ ---> $\"f%28x%29%3E0\"$ .
\n" ); document.write( "To find maxima and minima, we need to calculate the derivative:
\n" ); document.write( " $\"f%28x%29=x%2B3%2B8%2F%28x-3%29\"$ , so

.
\n" ); document.write( " $\"%22f+%27+%28+x+%29%22=0\"$ ---> $\"%28x-3%29%5E2=8\"$ ---> $\"x=3+%2B-+sqrt%288%29=x=3+%2B-+2sqrt%282%29\"$ .
\n" ); document.write( "Approximate values are $\"3-sqrt%288%29=0.17\"$ and $\"3%2Bsqrt%288%29=5.83\"$ .
\n" ); document.write( "So $\"f%283-sqrt%288%29%29=about%280.17%5E2-1%29%2F%280.17-3%29=about0.34\"$ and
\n" ); document.write( " $\"f%283%2Bsqrt%288%29%29=about%285.83%5E2-1%29%2F%285.83-3%29=about11.66\"$ .
\n" ); document.write( "We can graph the asymptotes, extreme value points, and zeros:
\n" ); document.write( "

, and with all else we found about $\"f%28x%29\"$ we sketch like this:

\n" ); document.write( "
\n" ); document.write( "d. Solve the inequality
\n" ); document.write( " $\"%28x%5E2-1%29%2F%28x-3%29%3C1\"$
\n" ); document.write( "We could look at the graph and see that the solution is $\"x%3C3\"$ .
\n" ); document.write( "Without the graph,
\n" ); document.write( " $\"%28x%5E2-1%29%2F%28x-3%29%3C1\"$ <--> $\"%28x%5E2-1%29%2F%28x-3%29-1%3C0\"$ <--> $\"%28x%5E2-1%29%2F%28x-3%29-%28x-3%29%2F%28x-3%29%3C0\"$ <--> $\"%28x%5E2-x%2B2%29%2F%28x-3%29%3C0\"$
\n" ); document.write( "Since the denominator is always positive, the solution is $\"x-3%3C0\"$ <--> $\"x%3C3\"$ .\r
\n" ); document.write( "\n" ); document.write( " \n" ); document.write( "

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