find domain, range, degree, x-intercept, y-intercept, vertical & horizontal asymtote, & holes

Question

the function is x2+4x divided by 4x+4

Answer 1

(x²+4x)/(4x+4).

DOMAIN

The denominator is zero when x=-1, so the domain is all values of x except x=-1 (asymptote).

Domain is (-∞,-1), (-1,∞).

RANGE

(-∞,∞)

DEGREE

The function is not a polynomial but its degree is 1, because the denominator divides into the numerator to produce a quotient in which the highest power of x is 1.

INTERCEPTS

When y=0, x²+4x=0, x(x+4)=0 so the x intercepts are 0 and -4.

When x=0, y=0, so the y-intercept is 0.

ASYMPTOTES

Vertical asymptote is x=-1. There is no horizontal asymptote but there is a sloping asymptote y=¼x. This is found by ignoring all but the highest power of x in the denominator and numerator:

If X is a very large value, (X²+4X)/(4X+4)≈X²/4X=X/4.

HOLES

None. There are no discontinuities apart from the asymptotes.