Question

Given f of x is equal to the quantity x plus 4 end quantity divided by the quantity x squared minus 3x minus 28 end quantity, which of the following is true?

f(x) is positive for all x > –4
f(x) is negative for all x > –4
f(x) is positive for all x < 7
f(x) is negative for all x < 7

Answer 1

`\( f(x) \) is positive for all \( x < 7 \).`

The function
`\( f(x) = (x + 4)/(x^2 - 3x - 28) \)` can be analyzed to determine its behavior across different ranges of x-values. To find where the function is positive or negative, it's essential to examine its factors and critical points.

Firstly, let's factor the denominator of the function to understand its behavior. The denominator
`\( x^2 - 3x - 28 \)`factors to
`\((x - 7)(x + 4)\).` This means the function has vertical asymptotes at
`\( x = 7 \)`and
`\( x = -4 \)`.

Now, consider the signs of the function in the intervals created by these critical points. When
`\( x < -4 \)`, both the numerator and denominator are negative, resulting in a positive
`\( f(x) \)` (negative divided by negative equals positive). Between
`\( -4 < x < 7 \)`, the numerator is positive while the denominator remains negative, making
`\( f(x) \)` negative (positive divided by negative equals negative). Finally, for
`\( x > 7 \)`, both the numerator and denominator are positive, leading to a positive
`\( f(x) \)` (positive divided by positive equals positive).

Given the options provided,
`\( f(x) \)` is positive for all ( x < 7 ). Therefore, the correct statement among the options is "f(x) is positive for all x < 7".