Question

which statement is true about the discontinuities of the function \\(f(x)\\)?

\\f(x) = \frac{x-5}{3x^2-17x-28}\\

• there are holes at \\(x = 7\\) and \\(x = -\frac{4}{3}\\).
• there are asymptotes at \\(x = 7\\) and \\(x = -\frac{4}{3}\\).
• there are asymptotes at \\(x = -7\\) and \\(x = \frac{4}{3}\\).
• there are holes at \\((-7, 0)\\) and \\(x = \frac{4}{3}\\).

Explanation:

Factor the denominator

Using the Polynomial Factoring knowledge point

$$ 3x^2 - 17x - 28 = (3x + 4)(x - 7) $$

Find the domain and discontinuities

Using the Holes in Rational Graphs and Vertical Asymptotes knowledge points

$$
LATEXBLOCK0
$$

Determine the type of discontinuities

Using the Vertical Asymptotes and Holes in Rational Graphs knowledge points

$$
LATEXBLOCK1
$$

Answer:

• There are holes at \(x = 7\) and \(x = -\frac{4}{3}\).
• There are asymptotes at \(x = 7\) and \(x = -\frac{4}{3}\). (Correct answer)
• There are asymptotes at \(x = -7\) and \(x = \frac{4}{3}\).
• There are holes at \((-7, 0)\) and \(x = \frac{4}{3}\).