Given The Function $f:\{2 \pi N \mid N \in \mathbb{Z} \} \rightarrow \{0\} ; F(x)=\sin X$, Check Which One(s) Of The Properties It Has.A. Surjective B. Injective C. Strictly Increasing D. Decreasing E. Strictly Decreasing F. Increasing

The sine function, denoted as $f(x) = \sin x$, is a fundamental function in mathematics, particularly in trigonometry and calculus. In this article, we will explore the properties of the sine function, specifically for the domain $\{2 \pi n \mid n \in \mathbb{Z} \}$ and range $\{0\}$. We will examine each of the given options to determine which properties the sine function satisfies.

Domain and Range

Before we dive into the properties of the sine function, let's clarify its domain and range. The domain of the sine function is the set of all real numbers, but in this case, we are considering the domain $\{2 \pi n \mid n \in \mathbb{Z} \}$. This means that the sine function is only defined at integer multiples of $2 \pi$. The range of the sine function is the set of all real numbers between -1 and 1, inclusive. However, in this case, the range is restricted to $\{0\}$.

Surjective Property

A function is said to be surjective if every element in the range is the image of at least one element in the domain. In other words, a function is surjective if every possible output value is actually reached by the function.

For the sine function, we need to determine if every element in the range $\{0\}$ is the image of at least one element in the domain $\{2 \pi n \mid n \in \mathbb{Z} \}$. Since the range is restricted to $\{0\}$, we only need to check if there exists an element in the domain that maps to 0.

The sine function has a period of $2 \pi$, which means that the function repeats itself every $2 \pi$ units. Therefore, for any integer multiple of $2 \pi$, the sine function will evaluate to 0. Specifically, we have:

$$f(2 \pi n) = \sin (2 \pi n) = 0$$

for all integers $n$. This shows that every element in the range $\{0\}$ is the image of at least one element in the domain $\{2 \pi n \mid n \in \mathbb{Z} \}$. Therefore, the sine function is surjective.

Injective Property

A function is said to be injective if every element in the domain maps to a unique element in the range. In other words, a function is injective if no two distinct elements in the domain map to the same element in the range.

For the sine function, we need to determine if every element in the domain $\{2 \pi n \mid n \in \mathbb{Z} \}$ maps to a unique element in the range $\{0\}$. Since the range is restricted to $\{0\}$, we only need to check if different elements in the domain map to the same element in the range.

However, we have already shown that the sine function is periodic with period $2 \pi$. This means that different elements in the domain can map to the same element in the range. Specifically, we have:

$$f(2 \pi n) = f(2 \pi (n + 1)) = 0$$

for all integers $n$. This shows that different elements in the domain can map to the same element in the range. Therefore, the sine function is not injective.

Strictly Increasing Property

A function is said to be strictly increasing if the function value increases as the input value increases. In other words, a function is strictly increasing if for any two elements $x_1$ and $x_2$ in the domain, we have:

$$x_1 < x_2 \implies f(x_1) < f(x_2)$$

For the sine function, we need to determine if the function value increases as the input value increases. However, since the range is restricted to $\{0\}$, we only need to check if the function value increases as the input value increases within the domain $\{2 \pi n \mid n \in \mathbb{Z} \}$.

However, we have already shown that the sine function is periodic with period $2 \pi$. This means that the function value does not increase as the input value increases. Specifically, we have:

$$f(2 \pi n) = f(2 \pi (n + 1)) = 0$$

for all integers $n$. This shows that the function value does not increase as the input value increases. Therefore, the sine function is not strictly increasing.

Decreasing Property

A function is said to be decreasing if the function value decreases as the input value increases. In other words, a function is decreasing if for any two elements $x_1$ and $x_2$ in the domain, we have:

$$x_1 < x_2 \implies f(x_1) > f(x_2)$$

For the sine function, we need to determine if the function value decreases as the input value increases. However, since the range is restricted to $\{0\}$, we only need to check if the function value decreases as the input value increases within the domain $\{2 \pi n \mid n \in \mathbb{Z} \}$.

However, we have already shown that the sine function is periodic with period $2 \pi$. This means that the function value does not decrease as the input value increases. Specifically, we have:

$$f(2 \pi n) = f(2 \pi (n + 1)) = 0$$

for all integers $n$. This shows that the function value does not decrease as the input value increases. Therefore, the sine function is not decreasing.

Strictly Decreasing Property

A function is said to be strictly decreasing if the function value decreases as the input value increases. In other words, a function is strictly decreasing if for any two elements $x_1$ and $x_2$ in the domain, we have:

$$x_1 < x_2 \implies f(x_1) > f(x_2)$$

For the sine function, we need to determine if the function value decreases as the input value increases. However, since the range is restricted to $\{0\}$, we only need to check if the function value decreases as the input value increases within the domain $\{2 \pi n \mid n \in \mathbb{Z} \}$.

However, we have already shown that the sine function is periodic with period $2 \pi$. This means that the function value does not decrease as the input value increases. Specifically, we have:

$$f(2 \pi n) = f(2 \pi (n + 1)) = 0$$

for all integers $n$. This shows that the function value does not decrease as the input value increases. Therefore, the sine function is not strictly decreasing.

Increasing Property

A function is said to be increasing if the function value increases as the input value increases. In other words, a function is increasing if for any two elements $x_1$ and $x_2$ in the domain, we have:

$$x_1 < x_2 \implies f(x_1) < f(x_2)$$

For the sine function, we need to determine if the function value increases as the input value increases. However, since the range is restricted to $\{0\}$, we only need to check if the function value increases as the input value increases within the domain $\{2 \pi n \mid n \in \mathbb{Z} \}$.

However, we have already shown that the sine function is periodic with period $2 \pi$. This means that the function value does not increase as the input value increases. Specifically, we have:

$$f(2 \pi n) = f(2 \pi (n + 1)) = 0$$

for all integers $n$. This shows that the function value does not increase as the input value increases. Therefore, the sine function is not increasing.

Conclusion

In conclusion, we have examined the properties of the sine function, specifically for the domain $\{2 \pi n \mid n \in \mathbb{Z} \}$ and range $\{0\}$. We have shown that the sine function is surjective, but not injective, strictly increasing, decreasing, strictly decreasing, or increasing. Therefore, the sine function satisfies only one of the given properties.

References

• [1] "Trigonometry" by Michael Corral
• [2] "Calculus" by Michael Spivak
• [3] "Mathematics for Computer Science" by Eric Lehman, F Thomson Leighton, and Albert R Meyer
  Q&A: Properties of the Sine Function =====================================

In our previous article, we explored the properties of the sine function, specifically for the domain $\{2 \pi n \mid n \in \mathbb{Z} \}$ and range $\{0\}$. We showed that the sine function is surjective, but not injective, strictly increasing, decreasing, strictly decreasing, or increasing. In this article, we will answer some frequently asked questions about the properties of the sine function.

Q: What is the domain of the sine function?

A: The domain of the sine function is the set of all real numbers, but in this case, we are considering the domain $\{2 \pi n \mid n \in \mathbb{Z} \}$.

Q: What is the range of the sine function?

A: The range of the sine function is the set of all real numbers between -1 and 1, inclusive. However, in this case, the range is restricted to $\{0\}$.

Q: Is the sine function surjective?

A: Yes, the sine function is surjective. This means that every element in the range $\{0\}$ is the image of at least one element in the domain $\{2 \pi n \mid n \in \mathbb{Z} \}$.

Q: Is the sine function injective?

A: No, the sine function is not injective. This means that different elements in the domain can map to the same element in the range.

Q: Is the sine function strictly increasing?

A: No, the sine function is not strictly increasing. This means that the function value does not increase as the input value increases.

Q: Is the sine function decreasing?

A: No, the sine function is not decreasing. This means that the function value does not decrease as the input value increases.

Q: Is the sine function strictly decreasing?

A: No, the sine function is not strictly decreasing. This means that the function value does not decrease as the input value increases.

Q: Is the sine function increasing?

A: No, the sine function is not increasing. This means that the function value does not increase as the input value increases.

Q: Why is the sine function not injective?

A: The sine function is not injective because it is periodic with period $2 \pi$. This means that different elements in the domain can map to the same element in the range.

Q: Why is the sine function not strictly increasing, decreasing, strictly decreasing, or increasing?

A: The sine function is not strictly increasing, decreasing, strictly decreasing, or increasing because it is periodic with period $2 \pi$. This means that the function value does not change as the input value increases or decreases.

Q: What are some real-world applications of the sine function?

A: The sine function has many real-world applications, including:

• Modeling periodic phenomena, such as the motion of a pendulum or the tides
• Describing the behavior of waves, such as sound or light waves
• Calculating distances and angles in navigation and surveying
• Modeling population growth and decline in biology and economics

Conclusion

In conclusion, we have answered some frequently asked questions about the properties of the sine function. We have shown that the sine function is surjective, but not injective, strictly increasing, decreasing, strictly decreasing, or increasing. We have also discussed some real-world applications of the sine function.