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
