Question

which of the following statements are true of this rational function? check all of the boxes that apply.

\\f(x) = \frac{(x + a)(x + b)}{x^2 + ax}\\

• there is a removable discontinuity at \\(x = -b\\).
• there is a vertical asymptote at \\(x = 0\\).
• there are no removable discontinuities.
• there is a vertical asymptote at \\(x = -a\\).
• there is a removable discontinuity at \\(x = -a\\).

Explanation:

Factor the denominator

Using the Rational Functions knowledge point

$$ f(x) = \frac{(x + a)(x + b)}{x(x + a)} $$

Identify removable discontinuities

Using the Removable Discontinuities knowledge point

$$
LATEXBLOCK0
$$

Identify vertical asymptotes

Using the Vertical Asymptotes knowledge point

$$
LATEXBLOCK1
$$

Evaluate the given statements

Using the Rational Functions knowledge point

$$
LATEXBLOCK2
$$

Answer:

• There is a removable discontinuity at \(x = -b\).
• There is a vertical asymptote at \(x = 0\). (Correct answer)
• There are no removable discontinuities.
• There is a vertical asymptote at \(x = -a\).
• There is a removable discontinuity at \(x = -a\). (Correct answer)