How do you find the domain, identify any horizontal, vertical, and slant (if possible) asymptotes and identify holes, x-intercepts, and y-intercepts for #(x^2-1)/(x^2+4)#?

1 Answer
Nov 16, 2017

Answer:

Domain = #RR in NN#
H.A. = 1
V.A. = non
S.A. = non
No Hole
x-int = #(1,0)# #(-1,0)#
y-int = #(0,-1/4)#

Explanation:

. Domain is all real numbers

.When the degree of denominator is equal to the degree of nominator,
H.A.=#"nominator's leading coefficient"/"denominator's leading coefficient"# therefore, #1/1# = 1 = Horizontal asymptote. .

.For rational functions, V.A. are the undefined points (zeros of the
denominator) of the simplified function.
#(x^2 -1)/(x^2+4)# is true and does not have any undefined points therefore no Vertical Asymptote.

. There are no slant asymptote because there is no V.A.

.There are no holes because there are no common factors in the
nominator and the denominator

. #x#-intercept is a point on the graph where #(y=0)#
#(x^2-1)/(x^2+4)=0#
#x^2-1=0#
solve using quadratic formula
#x=-0+- sqrt((0^2-4*1(-1))/(2*1)# #=+ or - 1#
#x=1 or x=-1#

. #y#-intercept is a point on the graph where #(x=0)#
#y=(0-1)/(0+4)# subtract each number by #0#
#y=(-1)/4#
#y=-1/4#