Question

which of the following statements are true of this rational function? check all of the boxes that apply. f(x)=(x^2 + ax)/((x + a)(x + b)) there is a removable discontinuity at x=-a. there is a vertical asymptote at x=-a. there are no removable discontinuities. there is a vertical asymptote at x = 0. there is a removable discontinuity at x=-b.

Explanation:

Step1: Recall function - discontinuity concepts

A removable discontinuity occurs when a factor in the denominator can be canceled out with a factor in the numerator. A vertical asymptote occurs when a factor in the denominator cannot be canceled and makes the denominator zero.

Step2: Analyze the function \(f(x)=\frac{x^{2}+ax}{(x + a)(x + b)}\)

First, factor the numerator: \(x^{2}+ax=x(x + a)\). Then \(f(x)=\frac{x(x + a)}{(x + a)(x + b)}\), for \(x
eq - a\) and \(x
eq - b\).

Step3: Check for removable discontinuities

Since we can cancel out the factor \((x + a)\) (assuming \(a
eq b\)), there is a removable discontinuity at \(x=-a\).

Step4: Check for vertical asymptotes

The function has a vertical asymptote at \(x=-b\) because the factor \((x + b)\) in the denominator cannot be canceled out. And \(x = 0\) is not a point of discontinuity as it does not make the denominator zero.

Answer:

There is a removable discontinuity at \(x=-a\)