Answer :
To find the point of the given function that corresponds to the minimum value of its inverse function, let's break down the problem step-by-step.
First, let's understand that we are dealing with a function \( f \) defined by the following points:
[tex]\[ \begin{array}{|c|c|c|c|c|c|} \hline x & -10 & -20 & 0 & 5 & 8 \\ \hline f(x) & 3 & 8 & -2 & -4.5 & -6 \\ \hline \end{array} \][/tex]
### Step 1: Identify the minimum value of \( f(x) \)
We need to find the smallest value in the second row, which contains the values of \( f(x) \):
[tex]\[ 3, 8, -2, -4.5, -6 \][/tex]
Among these values, the minimum value is \(-6\).
### Step 2: Find the corresponding \( x \) value for the minimum \( f(x) \)
Now, we will look at the table to identify the \( x \) value that corresponds to \( f(x) = -6 \):
[tex]\[ \begin{array}{|c|c|c|c|c|c|} \hline x & -10 & -20 & 0 & 5 & 8 \\ \hline f(x) & 3 & 8 & -2 & -4.5 & -6 \\ \hline \end{array} \][/tex]
From the table, we see that \( f(8) = -6 \).
### Step 3: Determine the point for the function’s inverse
For the inverse function \( f^{-1} \), what pairs \((f(x), x)\). Given that the minimum \( f(x) \) is -6, the corresponding point on the inverse function will be \( (-6, 8) \).
Alternatively, we reverse the pair to refer back to the original function point it came from:
Given that \( f(8) = -6 \):
The original function point is \( (8, -6) \).
### Conclusion
Thus, the point [tex]\((8, -6)\)[/tex] in the given function corresponds to the minimum value of its inverse function. None of the points [tex]\((-20, 8)\)[/tex] or [tex]\((-10, 3)\)[/tex] correspond correctly here; the correct point is [tex]\((8, -6)\)[/tex].
First, let's understand that we are dealing with a function \( f \) defined by the following points:
[tex]\[ \begin{array}{|c|c|c|c|c|c|} \hline x & -10 & -20 & 0 & 5 & 8 \\ \hline f(x) & 3 & 8 & -2 & -4.5 & -6 \\ \hline \end{array} \][/tex]
### Step 1: Identify the minimum value of \( f(x) \)
We need to find the smallest value in the second row, which contains the values of \( f(x) \):
[tex]\[ 3, 8, -2, -4.5, -6 \][/tex]
Among these values, the minimum value is \(-6\).
### Step 2: Find the corresponding \( x \) value for the minimum \( f(x) \)
Now, we will look at the table to identify the \( x \) value that corresponds to \( f(x) = -6 \):
[tex]\[ \begin{array}{|c|c|c|c|c|c|} \hline x & -10 & -20 & 0 & 5 & 8 \\ \hline f(x) & 3 & 8 & -2 & -4.5 & -6 \\ \hline \end{array} \][/tex]
From the table, we see that \( f(8) = -6 \).
### Step 3: Determine the point for the function’s inverse
For the inverse function \( f^{-1} \), what pairs \((f(x), x)\). Given that the minimum \( f(x) \) is -6, the corresponding point on the inverse function will be \( (-6, 8) \).
Alternatively, we reverse the pair to refer back to the original function point it came from:
Given that \( f(8) = -6 \):
The original function point is \( (8, -6) \).
### Conclusion
Thus, the point [tex]\((8, -6)\)[/tex] in the given function corresponds to the minimum value of its inverse function. None of the points [tex]\((-20, 8)\)[/tex] or [tex]\((-10, 3)\)[/tex] correspond correctly here; the correct point is [tex]\((8, -6)\)[/tex].
