Looking for reliable answers? Westonci.ca is the ultimate Q&A platform where experts share their knowledge on various topics. Our platform provides a seamless experience for finding reliable answers from a network of experienced professionals. Our platform provides a seamless experience for finding reliable answers from a network of experienced professionals.
Sagot :
Given the function [tex]\( g \)[/tex], we are provided with the following values for [tex]\( x \)[/tex] and [tex]\( g(x) \)[/tex]:
[tex]\[ \begin{array}{|c|c|c|c|c|c|} \hline x & -8 & -4 & 0 & 4 & 8 \\ \hline g(x) & 2 & 3 & 4 & 5 & 6 \\ \hline \end{array} \][/tex]
We need to determine which of the following given functions could be the inverse of [tex]\( g \)[/tex].
---
To find the inverse of [tex]\( g \)[/tex], we need a function that reverses the outputs back into the original inputs. Mathematically, if [tex]\( g(a) = b \)[/tex], then [tex]\( g^{-1}(b) = a \)[/tex].
We will check each option to identify which one satisfies this condition:
### Option A:
[tex]\[ \begin{array}{|c|c|c|c|c|c|} \hline x & 2 & 3 & 4 & 5 & 6 \\ \hline k(x) & -8 & -4 & 0 & 4 & 8 \\ \hline \end{array} \][/tex]
For [tex]\( k \)[/tex] to be the inverse of [tex]\( g \)[/tex]:
- [tex]\( g(-8) = 2 \)[/tex] implies [tex]\( k(2) = -8 \)[/tex]
- [tex]\( g(-4) = 3 \)[/tex] implies [tex]\( k(3) = -4 \)[/tex]
- [tex]\( g(0) = 4 \)[/tex] implies [tex]\( k(4) = 0 \)[/tex]
- [tex]\( g(4) = 5 \)[/tex] implies [tex]\( k(5) = 4 \)[/tex]
- [tex]\( g(8) = 6 \)[/tex] implies [tex]\( k(6) = 8 \)[/tex]
Clearly, [tex]\( k(x) \)[/tex] matches these requirements exactly. Therefore, option A is a valid choice.
### Checking Other Options for Completeness
#### Option B:
[tex]\[ \begin{array}{|c|c|c|c|c|c|} \hline x & -8 & -4 & 0 & 4 & 8 \\ \hline j(x) & -2 & -3 & -4 & 5 & 6 \\ \hline \end{array} \][/tex]
- [tex]\( g(-8) = 2 \)[/tex] so we need [tex]\( j(2) = -8 \)[/tex]
- [tex]\( g(-4) = 3 \)[/tex] so we need [tex]\( j(3) = -4 \)[/tex]
- [tex]\( g(0) = 4 \)[/tex] so we need [tex]\( j(4) = 0 \)[/tex]
- [tex]\( g(4) = 5 \)[/tex] so we need [tex]\( j(5) = 4 \)[/tex]
- [tex]\( g(8) = 6 \)[/tex] so we need [tex]\( j(6) = 8 \)[/tex]
These requirements are not matched by [tex]\( j(x) \)[/tex], so option B is not valid.
#### Option C:
[tex]\[ \begin{array}{|c|c|c|c|c|c|} \hline x & 2 & 3 & 0 & 4 & 8 \\ \hline m(x) & 8 & 4 & -4 & -5 & -6 \\ \hline \end{array} \][/tex]
The [tex]\( x \)[/tex] values don't match the outputs of [tex]\( g(x) \)[/tex] as inputs adequately, so [tex]\( m \)[/tex] cannot be the inverse.
#### Option D:
[tex]\[ \begin{array}{|c|c|c|c|c|c|} \hline x & -8 & -4 & 0 & 4 & 8 \\ \hline h(x) & -2 & -3 & -4 & -5 & -6 \\ \hline \end{array} \][/tex]
Here, [tex]\( h(x) \)[/tex] does not match the patterns needed to reverse [tex]\( g(x) \)[/tex], so option D is not valid.
---
Thus, the function that could be the inverse of [tex]\( g \)[/tex] is:
[tex]\[ \boxed{\text{A}} \][/tex]
[tex]\[ \begin{array}{|c|c|c|c|c|c|} \hline x & -8 & -4 & 0 & 4 & 8 \\ \hline g(x) & 2 & 3 & 4 & 5 & 6 \\ \hline \end{array} \][/tex]
We need to determine which of the following given functions could be the inverse of [tex]\( g \)[/tex].
---
To find the inverse of [tex]\( g \)[/tex], we need a function that reverses the outputs back into the original inputs. Mathematically, if [tex]\( g(a) = b \)[/tex], then [tex]\( g^{-1}(b) = a \)[/tex].
We will check each option to identify which one satisfies this condition:
### Option A:
[tex]\[ \begin{array}{|c|c|c|c|c|c|} \hline x & 2 & 3 & 4 & 5 & 6 \\ \hline k(x) & -8 & -4 & 0 & 4 & 8 \\ \hline \end{array} \][/tex]
For [tex]\( k \)[/tex] to be the inverse of [tex]\( g \)[/tex]:
- [tex]\( g(-8) = 2 \)[/tex] implies [tex]\( k(2) = -8 \)[/tex]
- [tex]\( g(-4) = 3 \)[/tex] implies [tex]\( k(3) = -4 \)[/tex]
- [tex]\( g(0) = 4 \)[/tex] implies [tex]\( k(4) = 0 \)[/tex]
- [tex]\( g(4) = 5 \)[/tex] implies [tex]\( k(5) = 4 \)[/tex]
- [tex]\( g(8) = 6 \)[/tex] implies [tex]\( k(6) = 8 \)[/tex]
Clearly, [tex]\( k(x) \)[/tex] matches these requirements exactly. Therefore, option A is a valid choice.
### Checking Other Options for Completeness
#### Option B:
[tex]\[ \begin{array}{|c|c|c|c|c|c|} \hline x & -8 & -4 & 0 & 4 & 8 \\ \hline j(x) & -2 & -3 & -4 & 5 & 6 \\ \hline \end{array} \][/tex]
- [tex]\( g(-8) = 2 \)[/tex] so we need [tex]\( j(2) = -8 \)[/tex]
- [tex]\( g(-4) = 3 \)[/tex] so we need [tex]\( j(3) = -4 \)[/tex]
- [tex]\( g(0) = 4 \)[/tex] so we need [tex]\( j(4) = 0 \)[/tex]
- [tex]\( g(4) = 5 \)[/tex] so we need [tex]\( j(5) = 4 \)[/tex]
- [tex]\( g(8) = 6 \)[/tex] so we need [tex]\( j(6) = 8 \)[/tex]
These requirements are not matched by [tex]\( j(x) \)[/tex], so option B is not valid.
#### Option C:
[tex]\[ \begin{array}{|c|c|c|c|c|c|} \hline x & 2 & 3 & 0 & 4 & 8 \\ \hline m(x) & 8 & 4 & -4 & -5 & -6 \\ \hline \end{array} \][/tex]
The [tex]\( x \)[/tex] values don't match the outputs of [tex]\( g(x) \)[/tex] as inputs adequately, so [tex]\( m \)[/tex] cannot be the inverse.
#### Option D:
[tex]\[ \begin{array}{|c|c|c|c|c|c|} \hline x & -8 & -4 & 0 & 4 & 8 \\ \hline h(x) & -2 & -3 & -4 & -5 & -6 \\ \hline \end{array} \][/tex]
Here, [tex]\( h(x) \)[/tex] does not match the patterns needed to reverse [tex]\( g(x) \)[/tex], so option D is not valid.
---
Thus, the function that could be the inverse of [tex]\( g \)[/tex] is:
[tex]\[ \boxed{\text{A}} \][/tex]
Thanks for stopping by. We are committed to providing the best answers for all your questions. See you again soon. We hope our answers were useful. Return anytime for more information and answers to any other questions you have. Keep exploring Westonci.ca for more insightful answers to your questions. We're here to help.
