Westonci.ca makes finding answers easy, with a community of experts ready to provide you with the information you seek. Join our platform to connect with experts ready to provide detailed answers to your questions in various areas. Experience the convenience of finding accurate answers to your questions from knowledgeable experts on our platform.
Sagot :
To determine whether the inverse of a function [tex]\( F(x) \)[/tex] is itself a function, we need to understand the properties of [tex]\( F(x) \)[/tex].
1. One-to-One Function (Injective):
- A function [tex]\( F(x) \)[/tex] is one-to-one if every value of [tex]\( y \)[/tex] (output) corresponds to exactly one value of [tex]\( x \)[/tex] (input). This means [tex]\( F(x_1) \neq F(x_2) \)[/tex] whenever [tex]\( x_1 \neq x_2 \)[/tex].
- If [tex]\( F(x) \)[/tex] is one-to-one, every unique input maps to a unique output ensuring that [tex]\( F(x) \)[/tex] has an inverse function.
2. Inverse of a Function:
- The inverse function [tex]\( F^{-1}(x) \)[/tex] essentially reverses the role of the input and output of [tex]\( F(x) \)[/tex].
- For [tex]\( F^{-1}(x) \)[/tex] to be a function, each element of the output of [tex]\( F(x) \)[/tex] must map back to only one element of the input of [tex]\( F(x) \)[/tex].
- This requirement will be met if [tex]\( F(x) \)[/tex] is one-to-one.
Given these points, if [tex]\( F(x) \)[/tex] is a one-to-one function, its inverse [tex]\( F^{-1}(x) \)[/tex] will also be a function. This ensures that each output [tex]\( y \)[/tex] in [tex]\( F(x) \)[/tex] maps to exactly one input [tex]\( x \)[/tex] in the inverse, thereby making the inverse a valid function.
Given this understanding, the correct answer to the question:
```
The inverse of F(x) is a function.
○ A. True
○ B. False
```
is:
○ A. True
1. One-to-One Function (Injective):
- A function [tex]\( F(x) \)[/tex] is one-to-one if every value of [tex]\( y \)[/tex] (output) corresponds to exactly one value of [tex]\( x \)[/tex] (input). This means [tex]\( F(x_1) \neq F(x_2) \)[/tex] whenever [tex]\( x_1 \neq x_2 \)[/tex].
- If [tex]\( F(x) \)[/tex] is one-to-one, every unique input maps to a unique output ensuring that [tex]\( F(x) \)[/tex] has an inverse function.
2. Inverse of a Function:
- The inverse function [tex]\( F^{-1}(x) \)[/tex] essentially reverses the role of the input and output of [tex]\( F(x) \)[/tex].
- For [tex]\( F^{-1}(x) \)[/tex] to be a function, each element of the output of [tex]\( F(x) \)[/tex] must map back to only one element of the input of [tex]\( F(x) \)[/tex].
- This requirement will be met if [tex]\( F(x) \)[/tex] is one-to-one.
Given these points, if [tex]\( F(x) \)[/tex] is a one-to-one function, its inverse [tex]\( F^{-1}(x) \)[/tex] will also be a function. This ensures that each output [tex]\( y \)[/tex] in [tex]\( F(x) \)[/tex] maps to exactly one input [tex]\( x \)[/tex] in the inverse, thereby making the inverse a valid function.
Given this understanding, the correct answer to the question:
```
The inverse of F(x) is a function.
○ A. True
○ B. False
```
is:
○ A. True
We hope this was helpful. Please come back whenever you need more information or answers to your queries. Thank you for choosing our platform. We're dedicated to providing the best answers for all your questions. Visit us again. Your questions are important to us at Westonci.ca. Visit again for expert answers and reliable information.
