Consider the function f(x) = −2x + 5 / x^2 − 5x − 6.
1. looking at the structure of the function, what information can you gather about the graph of f?
2. state the domain of f
3. determine the end behavior of f.
4. state the equations of any vertical and horizontal asymptotes on the graph of y = f(x).

Answer with Step-by-step explanation:

We are given that a function

1.

The given function is not define at x=-1 and x=6.

Therefore, at x=-1 and at x=6 limit of given function does not exist.

2.Domain of f=R-{-1,6}

3.

When x approaches infinity then function approach to zero.

4.Vertical asymptote :

Substitute denominator =0

Horizontal asymptote:

The degree of numerator=1

Degree of denominator=2

Degree of  numerator is less than the degree of denominator.Therefore,

Horizontal asymptote=0

1. It is a rational function.

2. .

3. End behaviors are shown below.

4. Vertical asymptotes : x=6 and x=-1; horizontal asymptote : y=0.

Step-by-step explanation:

The given function is

1.

A ration function is defined as

f(x)=p(x)/q(x)

The given function is in the form of p(x)/q(x).

Therefore, it is a rational function.

Find the factors of denominator.

Equate the denominator equal to 0.

Therefore, the function is not defined for x=6 and x=-1.

2.

The domain of given function is all real number except 6 and -1.

3.

End behavior of the function.

4.

To find the vertical asymptotes we need to find the zeroes of the denominator.

The vertical asymptotes of the function are x=6 and x=-1.

If degree of denominator is more than degree of numerator, then the horizontal asymptote is y=0.