Get the information you need from a community of experts on IDNLearn.com. Join our interactive community and get comprehensive, reliable answers to all your questions.
Sagot :
To determine which pairs of functions are inverses of each other, let’s analyze each given pair step-by-step:
### 1. Checking the functions:
[tex]\( f(x) = \{(-5,-9),(-3,-4),(0,1),(3,7),(6,13)\} \)[/tex]
[tex]\( g(x) = \{(-9,-5),(-4,-3),(1,0),(7,3),(13,6)\} \)[/tex]
To determine if [tex]\( f \)[/tex] and [tex]\( g \)[/tex] are inverses, we need to check if each pair [tex]\((x, y)\)[/tex] in [tex]\( f \)[/tex] corresponds to a pair [tex]\((y, x)\)[/tex] in [tex]\( g \)[/tex]:
- [tex]\((-5, -9)\)[/tex] in [tex]\( f \)[/tex] corresponds to [tex]\((-9, -5)\)[/tex] in [tex]\( g \)[/tex].
- [tex]\((-3, -4)\)[/tex] in [tex]\( f \)[/tex] corresponds to [tex]\((-4, -3)\)[/tex] in [tex]\( g \)[/tex].
- [tex]\((0, 1)\)[/tex] in [tex]\( f \)[/tex] corresponds to [tex]\((1, 0)\)[/tex] in [tex]\( g \)[/tex].
- [tex]\((3, 7)\)[/tex] in [tex]\( f \)[/tex] corresponds to [tex]\((7, 3)\)[/tex] in [tex]\( g \)[/tex].
- [tex]\((6, 13)\)[/tex] in [tex]\( f \)[/tex] corresponds to [tex]\((13, 6)\)[/tex] in [tex]\( g \)[/tex].
Every pair [tex]\((x, y)\)[/tex] in [tex]\( f \)[/tex] is matched with a pair [tex]\((y, x)\)[/tex] in [tex]\( g \)[/tex], implying that [tex]\( f \)[/tex] and [tex]\( g \)[/tex] are indeed inverses of each other.
### 2. Checking the linear functions:
[tex]\( f(x) = x + 7 \)[/tex]
[tex]\( g(x) = x - 7 \)[/tex]
To check if these functions are inverses, we need to evaluate their compositions:
- For [tex]\( g(f(x)) \)[/tex]:
[tex]\[ g(f(x)) = g(x + 7) = (x + 7) - 7 = x \][/tex]
- For [tex]\( f(g(x)) \)[/tex]:
[tex]\[ f(g(x)) = f(x - 7) = (x - 7) + 7 = x \][/tex]
Both compositions [tex]\( g(f(x)) \)[/tex] and [tex]\( f(g(x)) \)[/tex] yield [tex]\( x \)[/tex]. Therefore, [tex]\( f \)[/tex] and [tex]\( g \)[/tex] are inverses of each other.
### 3. Checking the tabular functions:
[tex]\[ \begin{array}{|c|c|c|c|c|c|} \hline \multirow{3}{*}{f(x)} & x & 2 & 3 & 4 & 5 \\ \cline { 1-6 } & y & 3 & 8 & 15 & 24 \\ \hline \hline \multirow{2}{*}{g(x)} & x & 5 & 4 & 3 & 2 \\ \cline { 2-6 } & y & 24 & 15 & 8 & 3 \\ \hline \end{array} \][/tex]
To determine if these functions are inverses, we need to check if each [tex]\((x, y)\)[/tex] in the representation of [tex]\( f \)[/tex] maps to a [tex]\((y, x)\)[/tex] in [tex]\( g \)[/tex]:
- [tex]\( (2, 3) \)[/tex] in [tex]\( f \)[/tex] should be [tex]\( (3, 2) \)[/tex] in [tex]\( g \)[/tex], but [tex]\( (3, 2) \)[/tex] does not exist in [tex]\( g \)[/tex].
- [tex]\( (3, 8) \)[/tex] in [tex]\( f \)[/tex] should be [tex]\( (8, 3) \)[/tex] in [tex]\( g \)[/tex], but [tex]\( (8, 3) \)[/tex] does not exist in [tex]\( g \)[/tex].
- Similarly, [tex]\( (4, 15) \)[/tex] in [tex]\( f \)[/tex] should correspond to [tex]\( (15, 4) \)[/tex] in [tex]\( g \)[/tex], but [tex]\( (15, 4) \)[/tex] does not exist in [tex]\( g \)[/tex].
The mapping does not occur as expected for inverses. Thus, these tabulated functions are not inverses of each other.
### Conclusion
The functions that are inverses of each other are:
1. [tex]\( f(x)=\{(-5,-9),(-3,-4),(0,1),(3,7),(6,13)\} \)[/tex] and [tex]\( g(x)=\{(-9,-5),(-4,-3),(1,0),(7,3),(13,6)\} \)[/tex]
2. [tex]\( f(x) = x + 7 \)[/tex] and [tex]\( g(x) = x - 7 \)[/tex]
### 1. Checking the functions:
[tex]\( f(x) = \{(-5,-9),(-3,-4),(0,1),(3,7),(6,13)\} \)[/tex]
[tex]\( g(x) = \{(-9,-5),(-4,-3),(1,0),(7,3),(13,6)\} \)[/tex]
To determine if [tex]\( f \)[/tex] and [tex]\( g \)[/tex] are inverses, we need to check if each pair [tex]\((x, y)\)[/tex] in [tex]\( f \)[/tex] corresponds to a pair [tex]\((y, x)\)[/tex] in [tex]\( g \)[/tex]:
- [tex]\((-5, -9)\)[/tex] in [tex]\( f \)[/tex] corresponds to [tex]\((-9, -5)\)[/tex] in [tex]\( g \)[/tex].
- [tex]\((-3, -4)\)[/tex] in [tex]\( f \)[/tex] corresponds to [tex]\((-4, -3)\)[/tex] in [tex]\( g \)[/tex].
- [tex]\((0, 1)\)[/tex] in [tex]\( f \)[/tex] corresponds to [tex]\((1, 0)\)[/tex] in [tex]\( g \)[/tex].
- [tex]\((3, 7)\)[/tex] in [tex]\( f \)[/tex] corresponds to [tex]\((7, 3)\)[/tex] in [tex]\( g \)[/tex].
- [tex]\((6, 13)\)[/tex] in [tex]\( f \)[/tex] corresponds to [tex]\((13, 6)\)[/tex] in [tex]\( g \)[/tex].
Every pair [tex]\((x, y)\)[/tex] in [tex]\( f \)[/tex] is matched with a pair [tex]\((y, x)\)[/tex] in [tex]\( g \)[/tex], implying that [tex]\( f \)[/tex] and [tex]\( g \)[/tex] are indeed inverses of each other.
### 2. Checking the linear functions:
[tex]\( f(x) = x + 7 \)[/tex]
[tex]\( g(x) = x - 7 \)[/tex]
To check if these functions are inverses, we need to evaluate their compositions:
- For [tex]\( g(f(x)) \)[/tex]:
[tex]\[ g(f(x)) = g(x + 7) = (x + 7) - 7 = x \][/tex]
- For [tex]\( f(g(x)) \)[/tex]:
[tex]\[ f(g(x)) = f(x - 7) = (x - 7) + 7 = x \][/tex]
Both compositions [tex]\( g(f(x)) \)[/tex] and [tex]\( f(g(x)) \)[/tex] yield [tex]\( x \)[/tex]. Therefore, [tex]\( f \)[/tex] and [tex]\( g \)[/tex] are inverses of each other.
### 3. Checking the tabular functions:
[tex]\[ \begin{array}{|c|c|c|c|c|c|} \hline \multirow{3}{*}{f(x)} & x & 2 & 3 & 4 & 5 \\ \cline { 1-6 } & y & 3 & 8 & 15 & 24 \\ \hline \hline \multirow{2}{*}{g(x)} & x & 5 & 4 & 3 & 2 \\ \cline { 2-6 } & y & 24 & 15 & 8 & 3 \\ \hline \end{array} \][/tex]
To determine if these functions are inverses, we need to check if each [tex]\((x, y)\)[/tex] in the representation of [tex]\( f \)[/tex] maps to a [tex]\((y, x)\)[/tex] in [tex]\( g \)[/tex]:
- [tex]\( (2, 3) \)[/tex] in [tex]\( f \)[/tex] should be [tex]\( (3, 2) \)[/tex] in [tex]\( g \)[/tex], but [tex]\( (3, 2) \)[/tex] does not exist in [tex]\( g \)[/tex].
- [tex]\( (3, 8) \)[/tex] in [tex]\( f \)[/tex] should be [tex]\( (8, 3) \)[/tex] in [tex]\( g \)[/tex], but [tex]\( (8, 3) \)[/tex] does not exist in [tex]\( g \)[/tex].
- Similarly, [tex]\( (4, 15) \)[/tex] in [tex]\( f \)[/tex] should correspond to [tex]\( (15, 4) \)[/tex] in [tex]\( g \)[/tex], but [tex]\( (15, 4) \)[/tex] does not exist in [tex]\( g \)[/tex].
The mapping does not occur as expected for inverses. Thus, these tabulated functions are not inverses of each other.
### Conclusion
The functions that are inverses of each other are:
1. [tex]\( f(x)=\{(-5,-9),(-3,-4),(0,1),(3,7),(6,13)\} \)[/tex] and [tex]\( g(x)=\{(-9,-5),(-4,-3),(1,0),(7,3),(13,6)\} \)[/tex]
2. [tex]\( f(x) = x + 7 \)[/tex] and [tex]\( g(x) = x - 7 \)[/tex]
Thank you for joining our conversation. Don't hesitate to return anytime to find answers to your questions. Let's continue sharing knowledge and experiences! Your search for answers ends at IDNLearn.com. Thank you for visiting, and we hope to assist you again soon.
