Question

(100 Points) Given f of x is equal to 10 divided by the quantity x squared minus 7x minus 30, which of the following is true?

f(x) is negative for all x < –3
f(x) is negative for all x > –3
f(x) is positive for all x < 10
f(x) is positive for all x > 10

Answer 1

Answer:

f(x) is positive for all x > 10

Step-by-step explanation:

Given function:

$f(x)=\dfrac{10}{x^2-7x-30}$

Asymptote

Asymptote: a line which the curve gets infinitely close to, but never touches.

Factor the denominator of the function to find the vertical asymptotes:

$\implies x^2-7x-30$

$\implies x^2-10x+3x-30$

$\implies x(x-10)+3(x-10)$

$\implies (x+3)(x-10)$

Therefore:

$f(x)=\dfrac{10}{(x+3)(x-10)}$

The function is undefined when the denominator is equal to zero.

Therefore, there are vertical asymptotes at x = -3  and  x = 10
and a horizontal asymptote at y = 0

f(x) is positive for (10, ∞)

f(x) is negative for (-3, 10)

f(x) is positive for (-∞, -3)

Answer 2

$\\ \rm\Rrightarrow y=\dfrac{10}{x^2-7x-10}$

If we factor

$\\ \rm\Rrightarrow y=\dfrac{10}{(x+3)(x-10)}$

Horizontal Asymptotes

• y=0 as there is no variable in numbers

Vertical asymptotes

solve the denominator for 0

• x=-3
• x=10

Henec option D is correct