Question

`f(x) = ((2x+3)(x-6))/((x+2)(x-1))`

Domain:
V.A:
Roots:
Y-int:
H.A:
Holes:
O.A:

Also, draw on the graph.

Answer 1

See below.

Explanation:

Domain is All Real x ; where x ≠ -2 , x ≠ 1.

Vertical asymptotes: x = -2 and x = 1.

Roots are the solution of (2x + 3)(x - 6) = 0. That is {-1.5, 6}.

Y-intercept is the value of y when x = 0:

= 2(0) + 3)(0 - 6) / ( 0 + 2)( 0 - 1)

= -18/-2 = 9. So it's (0, 9).

Horizontal asymptotes:

The limit of the function as x approaches infinity:

The terms in x^2 are 2x^2 in the numerator and x^2 in the denominator so the limit is +2.

There is an asymptote y = 2.

Holes:

No factors cancel out so there are no holes.

I'm not sure about O.A.