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

Answer:

  (a) positive for x < -5

Step-by-step explanation:

The sign of a rational function can be determined from the signs of its factors. The sign of the function will change when either the sign of the numerator or the sign of the denominator changes. The sign will change on either side of the zero of a linear factor with odd multiplicity.

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sign changes

The factorization of the given rational function is ....

  $f(x)=\dfrac{2}{x^2+3x-10}=\dfrac{2}{(x+5)(x-2)}$

Denominator zeros are ...

  x +5 = 0   ⇒   x = -5

  x -2 = 0   ⇒   x = 2

The sign of the function will change at x=-5 and at x=2.

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sign

The denominator factors will have different signs in the interval -5 < x < 2. That is the only interval in which the function value is negative. (Eliminates choices B and D.)

The denominator factors will have the same sign for x < -5 (both negative) or x > 2 (both positive). The function value will be positive in those intervals. (Eliminates choice C.)

  f(x) is positive for all x < -5.

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Additional comment

If a zero has even multiplicity, the sign of the function on either side of it is the same. The product of an even number of negatives has the same sign as the product of an even number of positives — positive.