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# Let f(x) = (x^2 - x - 2)/(x - 3) Please help me to find a solution. Thank You!

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This is the last thing I need help with! Please help me to find a solution. Thank You!  Apr 21, 2020

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f(x)  =  [ x2 - x - 2 ] / [ x - 3 ]

Divide each term by x:  f(x)  =  [ x2 /x  -  x / x  -  2 / x ] / [ x / x  -  3 / x ]   =  [ x - 1 - 2/x ] / [ 1 - 3/x ]

a)   As x --> infinity, this goes to  ( x - 1 ) / 1  =  x - 1  --->  infinity

b)   As x --> - infinity, this goes to  - infinity.

c)  Since the numerator has a degree one more than the denominator, it has a slant asymptote.

Doing long division, we get a quotient of  x + 2

Therefore, the equation of the slant asymptote is:  y  =  x + 2

d)  The denominator is  x - 3.

Therefore, there will be a vertical asymptote at  x  =  3.

e)  To find the x-intercepts, set y to 0 and solve:  0  =  [ x2 - x - 2 ] / [ x - 3 ]

0  =  x2 - x - 2

0  =  (x - 2)(x + 1)

Therefore, there will be x-intercepts at -1 and 2.

To find the y-intercept, set x to 0 and solve:  y  =  [ 02 - 0 - 2 ] / [ 0 - 3 ]

Therefore, there will be a y-intercept at 2/3.

There are no holes in the graph because there is no factor that is in both the numerator and denominator.

Graphing -- that's up to you ...

Apr 27, 2020