Answer:
a)
and
, b)
, c)
, d)
.
Step-by-step explanation:
a) After we read the statement carefully, we find that rational-polyomic function has the following characteristics:
1) A root of the polynomial at numerator is -2. (Removable discontinuity)
2) Roots of the polynomial at denominator are 1 and -2, respectively. (Vertical asymptote and removable discontinuity.
We analyze each polynomial by factorization and direct comparison to determine the values of
,
and
.
Denominator
i)
Given
ii)
Factorization
iii)
Compatibility with multiplication/Cancellative Property/Result
After a quick comparison, we conclude that
and 
b) The numerator is analyzed by applying the same approached of the previous item:
Numerator
i)
Given
ii)
Distributive Property
iii)
Distributive and Associative Properties/
/Result
As we know, this polynomial has
as one of its roots and therefore, the following identity must be met:
i)
Given
ii)
Compatibility with addition/Modulative property/Existence of additive inverse.
iii)
Definition of division/Existence of multiplicative inverse/Compatibility with multiplication/Modulative property/Result
The value of
is -10.
c) We can rewrite the rational function as:

After eliminating the removable discontinuity, the function becomes:

At
, we find that
is:
![f(-2) = -\frac{5}{2}\cdot \left[\frac{(-2)}{(-2)-1} \right]](https://tex.z-dn.net/?f=f%28-2%29%20%3D%20-%5Cfrac%7B5%7D%7B2%7D%5Ccdot%20%5Cleft%5B%5Cfrac%7B%28-2%29%7D%7B%28-2%29-1%7D%20%5Cright%5D)

d) The value of the horizontal asympote is equal to the limit of the rational function tending toward
. That is:
Given
Modulative Property
Existence of Multiplicative Inverse/Definition of Division

/
Limit properties/
, for 
The horizontal asymptote to the graph of f is
.