Question

which of the statements below best describes the periodic asymptotic behavior for the function (f(\theta) = \tan(3pi\theta))?

• the function (f) has vertical asymptotes at input values (\theta = 1 + \frac{1}{3}k), where (k) is an integer.
• the function (f) has vertical asymptotes at input values (\theta = 0 + \frac{1}{3}k), where (k) is an integer.
• the function (f) has vertical asymptotes at input values (\theta = \frac{1}{6} + \frac{1}{3}k), where (k) is an integer.
• the function (f) has vertical asymptotes at input values (\theta = \frac{1}{3} + \frac{1}{6}k), where (k) is an integer.

Explanation:

Identify the asymptote condition for the tangent function

Using the Tangent Function Period knowledge point

$$
LATEXBLOCK0
$$

Solve for the input variable

Using the Tangent Function Period knowledge point

$$
LATEXBLOCK1
$$

Match with the given multiple-choice options

Using the Tangent Function Period knowledge point

$$
LATEXBLOCK2
$$

Answer:

• The function f has vertical asymptotes at input values \(\theta = 1 + \frac{1}{3}k\), where k is an integer.
• The function f has vertical asymptotes at input values \(\theta = 0 + \frac{1}{3}k\), where k is an integer.
• The function f has vertical asymptotes at input values \(\theta = \frac{1}{6} + \frac{1}{3}k\), where k is an integer. (Correct answer)
• The function f has vertical asymptotes at input values \(\theta = \frac{1}{3} + \frac{1}{6}k\), where k is an integer.