The Periodic Function \[$ F \$\] Has A Period Of 4. The Function Is Increasing On The Input-value Interval \[$(0,4)\$\]. Which Of The Following Is True?A. \[$ F(9) \ \textless \ F(15) \$\], Because \[$ F(9) = F(1)
Understanding Periodic Functions
A periodic function is a function that repeats its values at regular intervals, known as the period. In this case, the function { f $}$ has a period of 4, meaning that its values repeat every 4 units of input. This is a fundamental concept in mathematics, particularly in calculus and analysis.
Properties of Periodic Functions
One of the key properties of periodic functions is that they have a repeating pattern. This means that if we know the value of the function at a certain point, we can determine the value of the function at other points that are a multiple of the period away. In this case, since the period is 4, we know that { f(1) = f(5) = f(9) = f(13) $}$, and so on.
Increasing on the Input-Value Interval
The problem states that the function is increasing on the input-value interval {(0,4)$}$. This means that as the input value increases, the output value of the function also increases. In other words, the function is monotonically increasing on this interval.
Analyzing the Options
Now, let's analyze the options given in the problem. We are asked to determine which of the following is true:
A. { f(9) \ \textless \ f(15) $}$, because { f(9) = f(1) B. [$ f(9) \ \textgreater \ f(15) $}$, because { f(9) = f(1) C. [$ f(9) = f(15) $}$, because { f(9) = f(1) D. [$ f(9) \ \textless \ f(15) $}$, because { f(9) = f(1) E. [$ f(9) \ \textgreater \ f(15) $}$, because { f(9) = f(1)
Option A
Let's start by analyzing option A. We are given that [$ f(9) \ \textless \ f(15) $}$, because { f(9) = f(1) This means that the value of the function at 9 is less than the value of the function at 15. However, since the function has a period of 4, we know that [$ f(9) = f(1) $}$. Therefore, we can conclude that { f(1) \ \textless \ f(15) $}$.
Option B
Now, let's analyze option B. We are given that { f(9) \ \textgreater \ f(15) $}$, because { f(9) = f(1) This means that the value of the function at 9 is greater than the value of the function at 15. However, since the function has a period of 4, we know that [$ f(9) = f(1) $}$. Therefore, we can conclude that { f(1) \ \textgreater \ f(15) $}$.
Option C
Next, let's analyze option C. We are given that { f(9) = f(15) $}$, because { f(9) = f(1) This means that the value of the function at 9 is equal to the value of the function at 15. However, since the function has a period of 4, we know that [$ f(9) = f(1) $}$. Therefore, we can conclude that { f(1) = f(15) $}$.
Option D
Now, let's analyze option D. We are given that { f(9) \ \textless \ f(15) $}$, because { f(9) = f(1) This means that the value of the function at 9 is less than the value of the function at 15. However, since the function has a period of 4, we know that [$ f(9) = f(1) $}$. Therefore, we can conclude that { f(1) \ \textless \ f(15) $}$.
Option E
Finally, let's analyze option E. We are given that { f(9) \ \textgreater \ f(15) $}$, because { f(9) = f(1) This means that the value of the function at 9 is greater than the value of the function at 15. However, since the function has a period of 4, we know that [$ f(9) = f(1) $}$. Therefore, we can conclude that { f(1) \ \textgreater \ f(15) $}$.
Conclusion
In conclusion, we have analyzed all the options and found that option C is the correct answer. The value of the function at 9 is equal to the value of the function at 15, because { f(9) = f(1) $}$. This is a direct consequence of the periodic nature of the function.
The Final Answer
Therefore, the final answer is:
C. { f(9) = f(15) $}$, because { f(9) = f(1)
Understanding the Periodic Function
A periodic function is a function that repeats its values at regular intervals, known as the period. In this case, the function [$ f $}$ has a period of 4, meaning that its values repeat every 4 units of input.
Properties of Periodic Functions
One of the key properties of periodic functions is that they have a repeating pattern. This means that if we know the value of the function at a certain point, we can determine the value of the function at other points that are a multiple of the period away.
Increasing on the Input-Value Interval
The problem states that the function is increasing on the input-value interval {(0,4)$}$. This means that as the input value increases, the output value of the function also increases.
Analyzing the Options
We are asked to determine which of the following is true:
A. { f(9) \ \textless \ f(15) $}$, because { f(9) = f(1) B. [$ f(9) \ \textgreater \ f(15) $}$, because { f(9) = f(1) C. [$ f(9) = f(15) $}$, because { f(9) = f(1) D. [$ f(9) \ \textless \ f(15) $}$, because { f(9) = f(1) E. [$ f(9) \ \textgreater \ f(15) $}$, because { f(9) = f(1)
Conclusion
In conclusion, we have analyzed all the options and found that option C is the correct answer. The value of the function at 9 is equal to the value of the function at 15, because [$ f(9) = f(1) $}$. This is a direct consequence of the periodic nature of the function.
The Final Answer
Therefore, the final answer is:
Q: What is a periodic function?
A: A periodic function is a function that repeats its values at regular intervals, known as the period. In this case, the function [$ f $}$ has a period of 4, meaning that its values repeat every 4 units of input.
Q: What are the properties of periodic functions?
A: One of the key properties of periodic functions is that they have a repeating pattern. This means that if we know the value of the function at a certain point, we can determine the value of the function at other points that are a multiple of the period away.
Q: What does it mean for a function to be increasing on an interval?
A: The problem states that the function is increasing on the input-value interval {(0,4)$}$. This means that as the input value increases, the output value of the function also increases.
Q: How do we determine the value of a periodic function at a point that is not in the interval?
A: Since the function has a period of 4, we can determine the value of the function at a point that is not in the interval by finding the equivalent point in the interval. For example, if we want to find the value of the function at 9, we can find the equivalent point in the interval by subtracting 4 from 9, which gives us 1. Therefore, { f(9) = f(1) $}$.
Q: What is the relationship between the value of a periodic function at two points that are a multiple of the period apart?
A: Since the function has a period of 4, the value of the function at two points that are a multiple of 4 apart is the same. For example, { f(1) = f(5) = f(9) = f(13) $}$.
Q: How do we determine which of the options is true?
A: We can determine which of the options is true by analyzing the properties of the periodic function and the given information. In this case, we can conclude that option C is the correct answer, because { f(9) = f(15) $}$, because { f(9) = f(1) $}$.
Q: What is the final answer?
A: The final answer is:
C. { f(9) = f(15) $}$, because { f(9) = f(1)
Frequently Asked Questions
- Q: What is a periodic function? A: A periodic function is a function that repeats its values at regular intervals, known as the period.
- Q: What are the properties of periodic functions? A: One of the key properties of periodic functions is that they have a repeating pattern.
- Q: What does it mean for a function to be increasing on an interval? A: The function is increasing on the interval if the output value increases as the input value increases.
- Q: How do we determine the value of a periodic function at a point that is not in the interval? A: We can determine the value of the function at a point that is not in the interval by finding the equivalent point in the interval.
Conclusion
In conclusion, we have analyzed the properties of periodic functions and determined which of the options is true. The final answer is:
C. [$ f(9) = f(15) $}$, because { f(9) = f(1)
The Final Answer
Therefore, the final answer is:
C. [$ f(9) = f(15) $} f(9) = f(1)