Question
A graph of y equals sine left-parenthesis start fraction 1 over x end fraction right-parenthesis is shown from x equals 0.01 to 0.05. The curve starts out oscillating between negative 1 and 1 very quickly and then gradually oscillates slower and slower.
Question
Use the graph to answer the question.
Look at the graph of f left parenthesis x right parenthesis equals sine Start Fraction 1 over x End Fraction. Is this function periodic? Why or why not?
(1 point)
Responses
Yes, it is periodic. The value of f left parenthesis x right parenthesis varies between negative 1 and 1 repeatedly.
Yes, it is periodic. The value of Image with alt text: f left parenthesis x right parenthesis varies between Image with alt text: negative 1 and 1 repeatedly.
Yes, it is periodic. It is a transformation of sin x.
Yes, it is periodic. It is a transformation of sin x .
No, it is not periodic. The function does not repeat on regular intervals.
No, it is not periodic. The function does not repeat on regular intervals.
No, it is not periodic. The function is a transformation of Start Fraction 1 over x End Fraction.
No, it is not periodic. The function is a transformation of Image with alt text: Start Fraction 1 over x End Fraction .
Question
Use the graph to answer the question.
Look at the graph of f left parenthesis x right parenthesis equals sine Start Fraction 1 over x End Fraction. Is this function periodic? Why or why not?
(1 point)
Responses
Yes, it is periodic. The value of f left parenthesis x right parenthesis varies between negative 1 and 1 repeatedly.
Yes, it is periodic. The value of Image with alt text: f left parenthesis x right parenthesis varies between Image with alt text: negative 1 and 1 repeatedly.
Yes, it is periodic. It is a transformation of sin x.
Yes, it is periodic. It is a transformation of sin x .
No, it is not periodic. The function does not repeat on regular intervals.
No, it is not periodic. The function does not repeat on regular intervals.
No, it is not periodic. The function is a transformation of Start Fraction 1 over x End Fraction.
No, it is not periodic. The function is a transformation of Image with alt text: Start Fraction 1 over x End Fraction .
Answers
GPT-4o mini
The function \( f(x) = \sin\left(\frac{1}{x}\right) \) is not periodic.
The reason is that for a function to be periodic, there must exist a positive number \( T \) such that \( f(x + T) = f(x) \) for all \( x \). In the case of \( \sin\left(\frac{1}{x}\right) \), as \( x \) approaches 0, the oscillations occur more and more rapidly, and there is no regular interval \( T \) that allows for the values of the function to repeat.
Thus, the correct response is:
**No, it is not periodic. The function does not repeat on regular intervals.**
The reason is that for a function to be periodic, there must exist a positive number \( T \) such that \( f(x + T) = f(x) \) for all \( x \). In the case of \( \sin\left(\frac{1}{x}\right) \), as \( x \) approaches 0, the oscillations occur more and more rapidly, and there is no regular interval \( T \) that allows for the values of the function to repeat.
Thus, the correct response is:
**No, it is not periodic. The function does not repeat on regular intervals.**