Check Your Understanding: Math 116B - Module 18Suppose F ( X ) = 7 − X F(x)=\sqrt{7-x} F ( X ) = 7 − X ​ . What Is The Domain Of F − 1 ( X F^{-1}(x F − 1 ( X ]?A) All Real Numbers Less Than Or Equal To 0. B) All Real Numbers Less Than Or Equal To 7. C) All Real Numbers Greater Than

Understanding the Domain of Inverse Functions

When dealing with inverse functions, it's essential to understand the concept of the domain and range. The domain of a function is the set of all possible input values for which the function is defined, while the range is the set of all possible output values. In the case of inverse functions, the domain of the inverse function is the range of the original function, and vice versa.

The Original Function

The original function is given as $f(x)=\sqrt{7-x}$. To find the domain of this function, we need to consider the values of $x$ for which the function is defined. Since the function involves a square root, the expression inside the square root must be non-negative. Therefore, we have:

$$7-x \geq 0$$

Solving for $x$, we get:

$$x \leq 7$$

This means that the domain of the original function $f(x)$ is all real numbers less than or equal to 7.

The Inverse Function

To find the domain of the inverse function $f^{-1}(x)$, we need to consider the values of $x$ for which the inverse function is defined. Since the inverse function is the reflection of the original function across the line $y=x$, the domain of the inverse function is the range of the original function.

Finding the Domain of the Inverse Function

To find the range of the original function, we need to consider the values of $f(x)$ for which the function is defined. Since the function involves a square root, the expression inside the square root must be non-negative. Therefore, we have:

$$7-x \geq 0$$

Solving for $f(x)$, we get:

$$f(x) \leq 7$$

This means that the range of the original function $f(x)$ is all real numbers less than or equal to 7.

Conclusion

Since the domain of the inverse function $f^{-1}(x)$ is the range of the original function $f(x)$, we can conclude that the domain of the inverse function $f^{-1}(x)$ is all real numbers less than or equal to 7.

The Correct Answer

The correct answer is B) All real numbers less than or equal to 7.

Discussion

The domain of the inverse function $f^{-1}(x)$ is the range of the original function $f(x)$. To find the domain of the inverse function, we need to consider the values of $x$ for which the inverse function is defined. Since the inverse function is the reflection of the original function across the line $y=x$, the domain of the inverse function is the range of the original function.

In this case, the original function is $f(x)=\sqrt{7-x}$. To find the range of this function, we need to consider the values of $f(x)$ for which the function is defined. Since the function involves a square root, the expression inside the square root must be non-negative. Therefore, we have:

$$7-x \geq 0$$

Solving for $f(x)$, we get:

$$f(x) \leq 7$$

This means that the range of the original function $f(x)$ is all real numbers less than or equal to 7.

Since the domain of the inverse function $f^{-1}(x)$ is the range of the original function $f(x)$, we can conclude that the domain of the inverse function $f^{-1}(x)$ is all real numbers less than or equal to 7.

Example

Suppose we have another function $g(x)=\sqrt{x-3}$. What is the domain of the inverse function $g^{-1}(x)$?

To find the domain of the inverse function, we need to consider the values of $x$ for which the inverse function is defined. Since the inverse function is the reflection of the original function across the line $y=x$, the domain of the inverse function is the range of the original function.

To find the range of the original function, we need to consider the values of $g(x)$ for which the function is defined. Since the function involves a square root, the expression inside the square root must be non-negative. Therefore, we have:

$$x-3 \geq 0$$

Solving for $g(x)$, we get:

$$g(x) \geq 3$$

This means that the range of the original function $g(x)$ is all real numbers greater than or equal to 3.

Since the domain of the inverse function $g^{-1}(x)$ is the range of the original function $g(x)$, we can conclude that the domain of the inverse function $g^{-1}(x)$ is all real numbers greater than or equal to 3.

Conclusion

In conclusion, the domain of the inverse function $f^{-1}(x)$ is the range of the original function $f(x)$. To find the domain of the inverse function, we need to consider the values of $x$ for which the inverse function is defined. Since the inverse function is the reflection of the original function across the line $y=x$, the domain of the inverse function is the range of the original function.

In this case, the original function is $f(x)=\sqrt{7-x}$. To find the range of this function, we need to consider the values of $f(x)$ for which the function is defined. Since the function involves a square root, the expression inside the square root must be non-negative. Therefore, we have:

$$7-x \geq 0$$

Solving for $f(x)$, we get:

$$f(x) \leq 7$$

This means that the range of the original function $f(x)$ is all real numbers less than or equal to 7.

Since the domain of the inverse function $f^{-1}(x)$ is the range of the original function $f(x)$, we can conclude that the domain of the inverse function $f^{-1}(x)$ is all real numbers less than or equal to 7.

Final Answer

The final answer is B) All real numbers less than or equal to 7.

Understanding the Domain of Inverse Functions

When dealing with inverse functions, it's essential to understand the concept of the domain and range. The domain of a function is the set of all possible input values for which the function is defined, while the range is the set of all possible output values. In the case of inverse functions, the domain of the inverse function is the range of the original function, and vice versa.

Q&A

Q: What is the domain of the inverse function $f^{-1}(x)$?

A: The domain of the inverse function $f^{-1}(x)$ is the range of the original function $f(x)$. In this case, the original function is $f(x)=\sqrt{7-x}$. To find the range of this function, we need to consider the values of $f(x)$ for which the function is defined. Since the function involves a square root, the expression inside the square root must be non-negative. Therefore, we have:

$$7-x \geq 0$$

Solving for $f(x)$, we get:

$$f(x) \leq 7$$

This means that the range of the original function $f(x)$ is all real numbers less than or equal to 7.

Q: How do I find the domain of the inverse function?

A: To find the domain of the inverse function, you need to consider the values of $x$ for which the inverse function is defined. Since the inverse function is the reflection of the original function across the line $y=x$, the domain of the inverse function is the range of the original function.

Q: What is the relationship between the domain of the original function and the domain of the inverse function?

A: The domain of the inverse function is the range of the original function, and vice versa. This means that if the original function has a domain of $[a,b]$, then the inverse function has a domain of $[a,b]$ as well.

Q: Can you give an example of finding the domain of the inverse function?

A: Suppose we have another function $g(x)=\sqrt{x-3}$. What is the domain of the inverse function $g^{-1}(x)$?

To find the domain of the inverse function, we need to consider the values of $x$ for which the inverse function is defined. Since the inverse function is the reflection of the original function across the line $y=x$, the domain of the inverse function is the range of the original function.

To find the range of the original function, we need to consider the values of $g(x)$ for which the function is defined. Since the function involves a square root, the expression inside the square root must be non-negative. Therefore, we have:

$$x-3 \geq 0$$

Solving for $g(x)$, we get:

$$g(x) \geq 3$$

This means that the range of the original function $g(x)$ is all real numbers greater than or equal to 3.

Since the domain of the inverse function $g^{-1}(x)$ is the range of the original function $g(x)$, we can conclude that the domain of the inverse function $g^{-1}(x)$ is all real numbers greater than or equal to 3.

Conclusion

In conclusion, the domain of the inverse function $f^{-1}(x)$ is the range of the original function $f(x)$. To find the domain of the inverse function, you need to consider the values of $x$ for which the inverse function is defined. Since the inverse function is the reflection of the original function across the line $y=x$, the domain of the inverse function is the range of the original function.

We hope this Q&A article has helped you understand the concept of the domain of inverse functions. If you have any further questions, please don't hesitate to ask.

Final Answer

The final answer is B) All real numbers less than or equal to 7.