Question

Given f of x is equal to 2 divided by the quantity x squared plus 3x minus 10 end quantity, which of the following is true? f(x) is positive for all x < –5 f(x) is negative for all x < –5 f(x) is positive for all x < 2 f(x) is negative for all x > 2

Answer 1

The function f(x) is positive for all x < -5 and for all x > 2, as the denominator's factors (x + 5)(x - 2) are both negative or both positive in these intervals respectively.

Step-by-step explanation:

To determine the sign of the function f(x) = = 2 / (x^2 + 3x - 10), we must look at the denominator, as the numerator is a positive constant. The denominator is a quadratic that can be factored as (x + 5)(x - 2). The quadratic will be positive when both factors are positive (x > 2) or both are negative (x < -5). It will be negative when the factors have opposite signs, which happens between the roots, for -5 < x < 2. Therefore, f(x) will be positive for all x < -5 and for all x > 2, and it will be negative for -5 < x < 2.