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Select the correct answer from each drop-down menu.

\begin{tabular}{|r|r|}
\hline
[tex]$x$[/tex] & [tex]$f(x)$[/tex] \\
\hline
-1 & 7 \\
\hline
1 & 6 \\
\hline
3 & 5 \\
\hline
4 & 1 \\
\hline
6 & -1 \\
\hline
\end{tabular}

The values in the table define the function [tex]$f(x)$[/tex]. Two points that lie on the graph of [tex]$f^{-1}(x)$[/tex] are [tex]$(7, -1)$[/tex] and \_\_\_.



Answer :

GinnyAnswer
To determine two points that lie on the graph of the inverse function [tex]\( f^{-1}(x) \)[/tex], we need to understand the relationship between [tex]\( f \)[/tex] and [tex]\( f^{-1} \)[/tex]. The inverse function [tex]\( f^{-1}(x) \)[/tex] essentially reverses the roles of the input and output of [tex]\( f(x) \)[/tex]. Specifically, if [tex]\( f(a) = b \)[/tex], then [tex]\( f^{-1}(b) = a \)[/tex].

Here’s the table for [tex]\( f(x) \)[/tex]:

[tex]\[ \begin{array}{|r|r|} \hline x & f(x) \\ \hline -1 & 7 \\ \hline 1 & 6 \\ \hline 3 & 5 \\ \hline 4 & 1 \\ \hline 6 & -1 \\ \hline \end{array} \][/tex]

1. For the first point, we observe that [tex]\( f(-1) = 7 \)[/tex]. Therefore, the point [tex]\( (7, -1) \)[/tex] lies on the graph of [tex]\( f^{-1}(x) \)[/tex].

2. For the second point, we need to pick another value from the function [tex]\( f(x) \)[/tex]. Notice that [tex]\( f(1) = 6 \)[/tex]. This means [tex]\( f^{-1}(6) = 1 \)[/tex], so the point [tex]\( (6, 1) \)[/tex] also lies on the graph of [tex]\( f^{-1}(x) \)[/tex].

Thus, the two points that lie on the graph of the inverse function [tex]\( f^{-1}(x) \)[/tex] are [tex]\( (7, -1) \)[/tex] and [tex]\( (6, 1) \)[/tex].

The correct answer is:
```
(7,-1) (6,1)
```

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