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

`\\ \rm\Rrightarrow y=(10)/(x^2-7x-10)`

If we factor

`\\ \rm\Rrightarrow y=(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

Answer 2

f(x) is positive for all x > 10

Explanation:

Given function:

`f(x)=(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)=(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)