Question

Consider the function f(x) = −2x + 5 / x^2 − 5x − 6.

Answer 1

Answer with Step-by-step explanation:

We are given that a function

$f(x)=\frac{-2x+5}{x^2-5x-6}$

1.$f(x)=\frac{-2x+5}{(x-6)(x+1)}$

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.$\lim_{x\rightarrow \infty}f(x)=\lim_{x\rightarrow \infty}\frac{x(-2+\frac{5}{x})}{x^2(1-\frac{5}{x}-\frac{6}{x^2})}=0$

When x approaches infinity then function approach to zero.

4.Vertical asymptote :

Substitute denominator =0

$(x-6)(x+1)=0$

$x+1=0\implies x=-1$

$x-6=0\implies x=6$

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

Answer 2

Answer:

1. It is a rational function.

2. $Domain=(-\infty,-1)\cup (-1,6)\cup (6,\infty)$.

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

$f(x)=\dfrac{-2x+5}{x^2-5x-6}$

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.

$f(x)=\dfrac{-2x+5}{x^2-6x+x-6}$

$f(x)=\dfrac{-2x+5}{x(x-6)+1(x-6)}$

$f(x)=\dfrac{-2x+5}{(x-6)(x+1)}$

Equate the denominator equal to 0.

$(x-6)(x+1)=0$

$x=6,-1$

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.

$Domain=(-\infty,-1)\cup (-1,6)\cup (6,\infty)$

3.

End behavior of the function.

$f(x)\rightarrow 0\text{ as } x\rightarrow -\infty$

$f(x)\rightarrow \infty\text{ as } x\rightarrow -1^-$

$f(x)\rightarrow -\infty\text{ as } x\rightarrow -1^+$

$f(x)\rightarrow \infty\text{ as } x\rightarrow 6^-$

$f(x)\rightarrow -\infty\text{ as } x\rightarrow 6^+$

$f(x)\rightarrow 0\text{ as } x\rightarrow \infty$

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.