The Function F F F Is Given By F ( X ) = 1 2 Sin ⁡ X F(x)=\frac{1}{2} \sin X F ( X ) = 2 1 ​ Sin X For − Π 2 ≤ X ≤ Π 2 -\frac{\pi}{2} \leq X \leq \frac{\pi}{2} − 2 Π ​ ≤ X ≤ 2 Π ​ . What Are The Domain And Range Of The Inverse Function Of F F F ?A. Domain: $\left[-\frac{1}{2},

Introduction

In mathematics, functions and their inverses play a crucial role in understanding various mathematical concepts. The given function $f(x)=\frac{1}{2} \sin x$ for $-\frac{\pi}{2} \leq x \leq \frac{\pi}{2}$ is a specific example of a trigonometric function. In this article, we will delve into the domain and range of the inverse function of $f$, which is denoted as $f^{-1}$.

Understanding the Original Function

Before we proceed to find the domain and range of the inverse function, let's understand the original function $f(x)=\frac{1}{2} \sin x$. The function is defined for the interval $-\frac{\pi}{2} \leq x \leq \frac{\pi}{2}$, which is a restricted domain for the sine function. The function is a scaled version of the sine function, with a coefficient of $\frac{1}{2}$.

The Inverse Function

To find the inverse function of $f$, we need to interchange the roles of $x$ and $y$ and solve for $y$. Let's denote the inverse function as $f^{-1}(x)$. We can start by writing the original function as $y = \frac{1}{2} \sin x$. Interchanging the roles of $x$ and $y$, we get $x = \frac{1}{2} \sin y$.

Solving for $y$

To solve for $y$, we can isolate the sine term by multiplying both sides of the equation by 2. This gives us $2x = \sin y$. Next, we can take the inverse sine of both sides to get $y = \sin^{-1}(2x)$.

Domain and Range of the Inverse Function

Now that we have the inverse function $f^{-1}(x) = \sin^{-1}(2x)$, let's analyze its domain and range.

Domain of the Inverse Function

The domain of the inverse function is the set of all possible input values for which the function is defined. In this case, the inverse function is defined for $-1 \leq x \leq 1$. This is because the sine function is defined for the interval $[-1, 1]$, and the coefficient of 2 scales this interval to $[-1, 1]$.

Range of the Inverse Function

The range of the inverse function is the set of all possible output values for which the function is defined. In this case, the inverse function is defined for the interval $[-\frac{\pi}{2}, \frac{\pi}{2}]$. This is because the inverse sine function is defined for the interval $[-\frac{\pi}{2}, \frac{\pi}{2}]$, and the coefficient of 2 scales this interval to $[-\frac{\pi}{2}, \frac{\pi}{2}]$.

Conclusion

In conclusion, the domain and range of the inverse function of $f(x)=\frac{1}{2} \sin x$ are $-1 \leq x \leq 1$ and $[-\frac{\pi}{2}, \frac{\pi}{2}]$, respectively. These results are obtained by analyzing the original function and its inverse, and understanding the properties of the sine and inverse sine functions.

References

• [1] Calculus by Michael Spivak
• [2] Trigonometry by I.M. Gelfand
• [3] Inverse Functions by Wolfram MathWorld

Discussion

Introduction

In our previous article, we explored the domain and range of the inverse function of $f(x)=\frac{1}{2} \sin x$. We analyzed the original function, found the inverse function, and determined the domain and range of the inverse function. In this article, we will address some common questions and provide additional insights on the topic.

Q&A

Q: What is the difference between the domain and range of a function and its inverse?

A: The domain of a function is the set of all possible input values for which the function is defined. The range of a function is the set of all possible output values for which the function is defined. When we find the inverse of a function, we interchange the roles of the input and output values. Therefore, the domain of the inverse function becomes the range of the original function, and vice versa.

Q: How do I determine the domain and range of an inverse function?

A: To determine the domain and range of an inverse function, you need to follow these steps:

1. Find the inverse function by interchanging the roles of the input and output values.
2. Analyze the inverse function to determine its domain and range.
3. Use the properties of the original function to determine the domain and range of the inverse function.

Q: What are some common mistakes to avoid when finding the domain and range of an inverse function?

A: Some common mistakes to avoid when finding the domain and range of an inverse function include:

• Not interchanging the roles of the input and output values when finding the inverse function.
• Not analyzing the inverse function to determine its domain and range.
• Not using the properties of the original function to determine the domain and range of the inverse function.

Q: Can you provide an example of a function and its inverse?

A: Let's consider the function $f(x) = 2x^2$. To find the inverse function, we can interchange the roles of the input and output values and solve for $x$. This gives us $x = \sqrt{\frac{y}{2}}$. Therefore, the inverse function is $f^{-1}(x) = \sqrt{\frac{x}{2}}$. The domain of the inverse function is $[0, \infty)$, and the range is $[0, \infty)$.

Q: How do I graph the inverse function?

A: To graph the inverse function, you can use the following steps:

1. Graph the original function.
2. Interchange the roles of the input and output values.
3. Reflect the graph of the original function across the line $y = x$.

Q: What are some real-world applications of inverse functions?

A: Inverse functions have many real-world applications, including:

• Modeling population growth and decline.
• Analyzing data and making predictions.
• Solving optimization problems.
• Finding the maximum or minimum value of a function.

Conclusion

In conclusion, the domain and range of an inverse function are determined by analyzing the original function and its inverse. By following the steps outlined in this article, you can determine the domain and range of an inverse function and apply this knowledge to real-world problems.

References

• [1] Calculus by Michael Spivak
• [2] Trigonometry by I.M. Gelfand
• [3] Inverse Functions by Wolfram MathWorld

Discussion

What are some other examples of functions and their inverses? How do you determine the domain and range of an inverse function? Share your thoughts and insights in the comments below!