Question

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)`?

Answer 1

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`