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Sagot :
To determine which function could be the inverse of the function [tex]\( f \)[/tex], we need to consider the definition of an inverse function. If [tex]\( f(x) \)[/tex] is a function, its inverse [tex]\( f^{-1}(x) \)[/tex] satisfies [tex]\( f(f^{-1}(x)) = x \)[/tex] and [tex]\( f^{-1}(f(x)) = x \)[/tex] for all [tex]\( x \)[/tex] in the domain of [tex]\( f \)[/tex] and [tex]\( f^{-1} \)[/tex], respectively.
Given the original 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]
Let's analyze each option to see if they could be the inverse of [tex]\( f \)[/tex].
### Option A:
[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]
For [tex]\( x = 1 \)[/tex], [tex]\( s(1) = -2 \)[/tex]. According to the table of [tex]\( f(x) \)[/tex], [tex]\( f^{-1}(-2) \)[/tex] should give us [tex]\( 1 \)[/tex]. However, [tex]\( f(-1) = -2 \)[/tex], thus [tex]\( f^{-1}(-2) = -1 \)[/tex], not [tex]\( 1 \)[/tex]. This option does not satisfy the requirement for the inverse function.
### Option B:
[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]
For [tex]\( x = 2 \)[/tex], [tex]\( r(2) = 1 \)[/tex], but in the table of [tex]\( f(x) \)[/tex], [tex]\( f(1) = 8 \)[/tex], so [tex]\( f^{-1}(8) = 1 \)[/tex] but not [tex]\( 2 \)[/tex]. For [tex]\( x = -3 \)[/tex], [tex]\( r(-3) = 0 \)[/tex]. For [tex]\( f(x) \)[/tex], there is no corresponding [tex]\( x \)[/tex] for [tex]\( f^{-1}(-3) \)[/tex]. This option is also incorrect as it does not correctly map the inverse relationships.
### Option C:
[tex]\[ \begin{array}{|c|c|c|c|c|} \hline x & -2 & 3 & 8 & 13 \\ \hline q(x) & -1 & 0 & 1 & 2 \\ \hline \end{array} \][/tex]
For [tex]\( x = -2 \)[/tex], [tex]\( q(-2) = -1 \)[/tex]. In the original function, [tex]\( f(-1) = -2 \)[/tex], which confirms [tex]\( f^{-1}(-2) = -1 \)[/tex]. For [tex]\( x = 3 \)[/tex], [tex]\( q(3) = 0 \)[/tex] and [tex]\( f(0) = 3 \)[/tex], verifying that [tex]\( f^{-1}(3) = 0 \)[/tex]. This pattern holds for each given pair. Thus, this option satisfies the conditions for being the inverse of [tex]\( f \)[/tex].
### Option D:
[tex]\[ \begin{array}{|c|c|c|c|c|} \hline x & -1 & 0 & 1 & 2 \\ \hline p(x) & 2 & -3 & -8 & -13 \\ \hline \end{array} \][/tex]
For [tex]\( x = -1 \)[/tex], [tex]\( p(-1) = 2 \)[/tex], but in the table of [tex]\( f(x) \)[/tex], [tex]\( f(2) = 13 \)[/tex], so [tex]\( f^{-1}(2) \)[/tex] does not match the inverse value of [tex]\( 2 \)[/tex]. This option does not satisfy the inverse property.
Thus, the correct function that acts as the inverse of [tex]\( f \)[/tex] is given by:
Option C.
[tex]\[ \begin{array}{|c|c|c|c|c|} \hline x & -2 & 3 & 8 & 13 \\ \hline q(x) & -1 & 0 & 1 & 2 \\ \hline \end{array} \][/tex]
So, the correct answer is:
[tex]\[ \boxed{3} \][/tex]
Given the original 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]
Let's analyze each option to see if they could be the inverse of [tex]\( f \)[/tex].
### Option A:
[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]
For [tex]\( x = 1 \)[/tex], [tex]\( s(1) = -2 \)[/tex]. According to the table of [tex]\( f(x) \)[/tex], [tex]\( f^{-1}(-2) \)[/tex] should give us [tex]\( 1 \)[/tex]. However, [tex]\( f(-1) = -2 \)[/tex], thus [tex]\( f^{-1}(-2) = -1 \)[/tex], not [tex]\( 1 \)[/tex]. This option does not satisfy the requirement for the inverse function.
### Option B:
[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]
For [tex]\( x = 2 \)[/tex], [tex]\( r(2) = 1 \)[/tex], but in the table of [tex]\( f(x) \)[/tex], [tex]\( f(1) = 8 \)[/tex], so [tex]\( f^{-1}(8) = 1 \)[/tex] but not [tex]\( 2 \)[/tex]. For [tex]\( x = -3 \)[/tex], [tex]\( r(-3) = 0 \)[/tex]. For [tex]\( f(x) \)[/tex], there is no corresponding [tex]\( x \)[/tex] for [tex]\( f^{-1}(-3) \)[/tex]. This option is also incorrect as it does not correctly map the inverse relationships.
### Option C:
[tex]\[ \begin{array}{|c|c|c|c|c|} \hline x & -2 & 3 & 8 & 13 \\ \hline q(x) & -1 & 0 & 1 & 2 \\ \hline \end{array} \][/tex]
For [tex]\( x = -2 \)[/tex], [tex]\( q(-2) = -1 \)[/tex]. In the original function, [tex]\( f(-1) = -2 \)[/tex], which confirms [tex]\( f^{-1}(-2) = -1 \)[/tex]. For [tex]\( x = 3 \)[/tex], [tex]\( q(3) = 0 \)[/tex] and [tex]\( f(0) = 3 \)[/tex], verifying that [tex]\( f^{-1}(3) = 0 \)[/tex]. This pattern holds for each given pair. Thus, this option satisfies the conditions for being the inverse of [tex]\( f \)[/tex].
### Option D:
[tex]\[ \begin{array}{|c|c|c|c|c|} \hline x & -1 & 0 & 1 & 2 \\ \hline p(x) & 2 & -3 & -8 & -13 \\ \hline \end{array} \][/tex]
For [tex]\( x = -1 \)[/tex], [tex]\( p(-1) = 2 \)[/tex], but in the table of [tex]\( f(x) \)[/tex], [tex]\( f(2) = 13 \)[/tex], so [tex]\( f^{-1}(2) \)[/tex] does not match the inverse value of [tex]\( 2 \)[/tex]. This option does not satisfy the inverse property.
Thus, the correct function that acts as the inverse of [tex]\( f \)[/tex] is given by:
Option C.
[tex]\[ \begin{array}{|c|c|c|c|c|} \hline x & -2 & 3 & 8 & 13 \\ \hline q(x) & -1 & 0 & 1 & 2 \\ \hline \end{array} \][/tex]
So, the correct answer is:
[tex]\[ \boxed{3} \][/tex]
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