Question 112972
The given expression is: y = f(x) = {{{ (x^2 + 5)/(x + 1)}}} 


The intercepts are given as follows:  


Y - intercept put x = 0  


==> y = (0 + 5)/(0 + 1) 


==> y = 5/1 


==> y = 5 


The x-intercepts are given when y = 0  


==> 0 = {{{(x^2 + 5)/(x + 1)}}} 


==> 0 = {{{x^2 + 5}}} 


==> x^2 = - 5 


==> {{{ x = i(sqrt(5))}}} (+ -) 



The graph looks this way: 


{{{ graph( 300, 300, -15, 15, -15, 15, (x^2 + 5)/(x + 1)) }}} 


The vertical and the horizontal asymptotes are found as follows: 



The vertical asymptotes come from the zeroes of the denominator, so I'll set the denominator equal to zero and solve.


x + 1 = 0 


==> x = -1 


Thus the vertical asymptotes.


Since the degree of the numerator is one greater than the degree of the denominator, I'll have a slant asymptote (not a horizontal one), and I'll find the slant asymptote by long division.


Thus by long dision we find that y = x + 1 is the slant asymptote for the given equation.