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Sagot :
To determine which function could be the inverse of function [tex]\( f \)[/tex], we need to verify if any of the given functions reverse the effect of [tex]\( f \)[/tex]. An inverse function [tex]\( f^{-1} \)[/tex] satisfies [tex]\( f(f^{-1}(x)) = x \)[/tex] for all values in the original function's domain. Let's analyze each option:
### Given function:
[tex]\[ \begin{array}{|c|c|c|c|c|} \hline x & -1 & 0 & 1 & 2 \\ \hline f(x) & -2 & 3 & 8 & 13 \\ \hline \end{array} \][/tex]
### Option A:
[tex]\[ \begin{array}{|c|c|c|c|c|} \hline x & 2 & -3 & -8 & -13 \\ \hline r(x) & 1 & 0 & -1 & -2 \\ \hline \end{array} \][/tex]
If [tex]\( r \)[/tex] were the inverse of [tex]\( f \)[/tex], then:
1. [tex]\( f(r(2)) = 2 \)[/tex]
2. [tex]\( f(r(-3)) = -3 \)[/tex]
3. [tex]\( f(r(-8)) = -8 \)[/tex]
4. [tex]\( f(r(-13)) = -13 \)[/tex]
Let's check:
1. [tex]\( r(2) = 1 \Rightarrow f(1) = 8 \neq 2 \)[/tex]
2. [tex]\( r(-3) = 0 \Rightarrow f(0) = 3 \neq -3 \)[/tex]
3. [tex]\( r(-8) = -1 \Rightarrow f(-1) = -2 \neq -8 \)[/tex]
4. [tex]\( r(-13) = -2 \Rightarrow f(2) = 13 \neq -13 \)[/tex]
These verifications show that Option A does not satisfy the conditions for an inverse function.
### Option B:
[tex]\[ \begin{array}{|c|c|c|c|c|} \hline x & 1 & 0 & -1 & -2 \\ \hline s(x) & -2 & 3 & 8 & 13 \\ \hline \end{array} \][/tex]
If [tex]\( s \)[/tex] were the inverse of [tex]\( f \)[/tex], then:
1. [tex]\( f(s(1)) = 1 \)[/tex]
2. [tex]\( f(s(0)) = 0 \)[/tex]
3. [tex]\( f(s(-1)) = -1 \)[/tex]
4. [tex]\( f(s(-2)) = -2 \)[/tex]
Let's check:
1. [tex]\( s(1) = -2 \Rightarrow f(-2) = -2 \neq 1 \)[/tex]
2. [tex]\( s(0) = 3 \Rightarrow f(3) \)[/tex] does not exist in original table.
3. [tex]\( s(-1) = 8 \Rightarrow f(8) \)[/tex] does not exist in original table.
4. [tex]\( s(-2) = 13 \Rightarrow f(13) \)[/tex] does not exist in original table.
These verifications show that Option B does not satisfy the conditions for an inverse function.
### Option C:
[tex]\[ \begin{array}{|c|c|c|c|c|} \hline x & -2 & 3 & 8 & 13 \\ \hline n(v) & -1 & \text{None} & 1 & \text{None} \\ \hline \end{array} \][/tex]
If [tex]\( n \)[/tex] were the inverse of [tex]\( f \)[/tex], then:
1. [tex]\( f(n(-2)) = -2 \)[/tex]
2. [tex]\( f(n(3)) = 3 \)[/tex]
3. [tex]\( f(n(8)) = 8 \)[/tex]
4. [tex]\( f(n(13)) = 13 \)[/tex]
Let's check:
1. [tex]\( n(-2) = -1 \Rightarrow f(-1) = -2 = -2 \)[/tex]
2. [tex]\( n(3) = \text{None} \Rightarrow \)[/tex] There is no verification needed if it does not exist.
3. [tex]\( n(8) = 1 \Rightarrow f(1) = 8 = 8 \)[/tex]
4. [tex]\( n(13) = \text{None} \Rightarrow \)[/tex] There is no verification needed if it does not exist.
Thus Option C meets the verification, therefore Option C is the inverse function.
This concludes that the correct answer is option:
[tex]\[ \boxed{2} \][/tex]
### Given function:
[tex]\[ \begin{array}{|c|c|c|c|c|} \hline x & -1 & 0 & 1 & 2 \\ \hline f(x) & -2 & 3 & 8 & 13 \\ \hline \end{array} \][/tex]
### Option A:
[tex]\[ \begin{array}{|c|c|c|c|c|} \hline x & 2 & -3 & -8 & -13 \\ \hline r(x) & 1 & 0 & -1 & -2 \\ \hline \end{array} \][/tex]
If [tex]\( r \)[/tex] were the inverse of [tex]\( f \)[/tex], then:
1. [tex]\( f(r(2)) = 2 \)[/tex]
2. [tex]\( f(r(-3)) = -3 \)[/tex]
3. [tex]\( f(r(-8)) = -8 \)[/tex]
4. [tex]\( f(r(-13)) = -13 \)[/tex]
Let's check:
1. [tex]\( r(2) = 1 \Rightarrow f(1) = 8 \neq 2 \)[/tex]
2. [tex]\( r(-3) = 0 \Rightarrow f(0) = 3 \neq -3 \)[/tex]
3. [tex]\( r(-8) = -1 \Rightarrow f(-1) = -2 \neq -8 \)[/tex]
4. [tex]\( r(-13) = -2 \Rightarrow f(2) = 13 \neq -13 \)[/tex]
These verifications show that Option A does not satisfy the conditions for an inverse function.
### Option B:
[tex]\[ \begin{array}{|c|c|c|c|c|} \hline x & 1 & 0 & -1 & -2 \\ \hline s(x) & -2 & 3 & 8 & 13 \\ \hline \end{array} \][/tex]
If [tex]\( s \)[/tex] were the inverse of [tex]\( f \)[/tex], then:
1. [tex]\( f(s(1)) = 1 \)[/tex]
2. [tex]\( f(s(0)) = 0 \)[/tex]
3. [tex]\( f(s(-1)) = -1 \)[/tex]
4. [tex]\( f(s(-2)) = -2 \)[/tex]
Let's check:
1. [tex]\( s(1) = -2 \Rightarrow f(-2) = -2 \neq 1 \)[/tex]
2. [tex]\( s(0) = 3 \Rightarrow f(3) \)[/tex] does not exist in original table.
3. [tex]\( s(-1) = 8 \Rightarrow f(8) \)[/tex] does not exist in original table.
4. [tex]\( s(-2) = 13 \Rightarrow f(13) \)[/tex] does not exist in original table.
These verifications show that Option B does not satisfy the conditions for an inverse function.
### Option C:
[tex]\[ \begin{array}{|c|c|c|c|c|} \hline x & -2 & 3 & 8 & 13 \\ \hline n(v) & -1 & \text{None} & 1 & \text{None} \\ \hline \end{array} \][/tex]
If [tex]\( n \)[/tex] were the inverse of [tex]\( f \)[/tex], then:
1. [tex]\( f(n(-2)) = -2 \)[/tex]
2. [tex]\( f(n(3)) = 3 \)[/tex]
3. [tex]\( f(n(8)) = 8 \)[/tex]
4. [tex]\( f(n(13)) = 13 \)[/tex]
Let's check:
1. [tex]\( n(-2) = -1 \Rightarrow f(-1) = -2 = -2 \)[/tex]
2. [tex]\( n(3) = \text{None} \Rightarrow \)[/tex] There is no verification needed if it does not exist.
3. [tex]\( n(8) = 1 \Rightarrow f(1) = 8 = 8 \)[/tex]
4. [tex]\( n(13) = \text{None} \Rightarrow \)[/tex] There is no verification needed if it does not exist.
Thus Option C meets the verification, therefore Option C is the inverse function.
This concludes that the correct answer is option:
[tex]\[ \boxed{2} \][/tex]
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