Question
Consider the function f left parenthesis x right parenthesis equals cotangent x. Which of the following are true? Select all that apply.
(1 point)
Responses
f left parenthesis x right parenthesis is an even function.
Image with alt text: f left parenthesis x right parenthesis is an even function.
f left parenthesis x right parenthesis has an asymptote at x = 0.
Image with alt text: f left parenthesis x right parenthesis has an asymptote at x = 0.
f left parenthesis x right parenthesis has a zero at x = 0.
Image with alt text: f left parenthesis x right parenthesis has a zero at x = 0.
f left parenthesis x right parenthesis has a period of π.
(1 point)
Responses
f left parenthesis x right parenthesis is an even function.
Image with alt text: f left parenthesis x right parenthesis is an even function.
f left parenthesis x right parenthesis has an asymptote at x = 0.
Image with alt text: f left parenthesis x right parenthesis has an asymptote at x = 0.
f left parenthesis x right parenthesis has a zero at x = 0.
Image with alt text: f left parenthesis x right parenthesis has a zero at x = 0.
f left parenthesis x right parenthesis has a period of π.
Answers
Answered by
GPT-4o mini
To analyze the function \( f(x) = \cot(x) \), we can evaluate each statement one by one:
1. **\( f(x) \) is an even function.**
A function \( f(x) \) is even if \( f(-x) = f(x) \) for all \( x \). For \( f(x) = \cot(x) \):
\[
f(-x) = \cot(-x) = -\cot(x).
\]
Since \( f(-x) \neq f(x) \), \( f(x) \) is not an even function. This statement is **false**.
2. **\( f(x) \) has an asymptote at \( x = 0 \).**
The cotangent function \( \cot(x) = \frac{\cos(x)}{\sin(x)} \) has vertical asymptotes where \( \sin(x) = 0 \), which occurs at \( x = n\pi \) for \( n \in \mathbb{Z} \). Thus, there is indeed a vertical asymptote at \( x = 0 \). This statement is **true**.
3. **\( f(x) \) has a zero at \( x = 0 \).**
To determine if \( f(x) \) has a zero at \( x = 0 \), we can check:
\[
f(0) = \cot(0) = \frac{\cos(0)}{\sin(0)}.
\]
Since \( \sin(0) = 0 \), \( f(0) \) is undefined and thus does not have a zero at \( x = 0 \). This statement is **false**.
4. **\( f(x) \) has a period of \( \pi \).**
A function \( f(x) \) is periodic with period \( T \) if \( f(x + T) = f(x) \) for all \( x \). For \( f(x) = \cot(x) \):
\[
f(x + \pi) = \cot(x + \pi) = \cot(x).
\]
This shows that \( f(x) \) is periodic with period \( \pi \). This statement is **true**.
In conclusion, the true statements about \( f(x) = \cot(x) \) are:
- \( f(x) \) has an asymptote at \( x = 0 \).
- \( f(x) \) has a period of \( \pi \).
1. **\( f(x) \) is an even function.**
A function \( f(x) \) is even if \( f(-x) = f(x) \) for all \( x \). For \( f(x) = \cot(x) \):
\[
f(-x) = \cot(-x) = -\cot(x).
\]
Since \( f(-x) \neq f(x) \), \( f(x) \) is not an even function. This statement is **false**.
2. **\( f(x) \) has an asymptote at \( x = 0 \).**
The cotangent function \( \cot(x) = \frac{\cos(x)}{\sin(x)} \) has vertical asymptotes where \( \sin(x) = 0 \), which occurs at \( x = n\pi \) for \( n \in \mathbb{Z} \). Thus, there is indeed a vertical asymptote at \( x = 0 \). This statement is **true**.
3. **\( f(x) \) has a zero at \( x = 0 \).**
To determine if \( f(x) \) has a zero at \( x = 0 \), we can check:
\[
f(0) = \cot(0) = \frac{\cos(0)}{\sin(0)}.
\]
Since \( \sin(0) = 0 \), \( f(0) \) is undefined and thus does not have a zero at \( x = 0 \). This statement is **false**.
4. **\( f(x) \) has a period of \( \pi \).**
A function \( f(x) \) is periodic with period \( T \) if \( f(x + T) = f(x) \) for all \( x \). For \( f(x) = \cot(x) \):
\[
f(x + \pi) = \cot(x + \pi) = \cot(x).
\]
This shows that \( f(x) \) is periodic with period \( \pi \). This statement is **true**.
In conclusion, the true statements about \( f(x) = \cot(x) \) are:
- \( f(x) \) has an asymptote at \( x = 0 \).
- \( f(x) \) has a period of \( \pi \).