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Comparing a Function and Its Inverse

Consider the tables created with Fahrenheit and Celsius temperatures. Table A represents the function that models Celsius temperature, [tex]C(F)[/tex], based on the given Fahrenheit temperature, [tex]F[/tex].

Table A:
\begin{tabular}{|c|c|c|c|}
\hline
[tex]F[/tex] & -4 & 5 & 23 \\
\hline
[tex]C(F)[/tex] & -20 & -15 & -5 \\
\hline
\end{tabular}

Which table could be used with Table A to verify that the function modeling Fahrenheit temperature, [tex]F(C)[/tex], based on a given Celsius temperature, [tex]C[/tex], is the inverse of [tex]C(F)[/tex]?

A.
\begin{tabular}{|c|c|c|c|}
\hline
[tex]C[/tex] & -20 & -15 & -5 \\
\hline
[tex]F(C)[/tex] & -4 & 5 & 23 \\
\hline
\end{tabular}

B.
\begin{tabular}{|c|c|c|c|}
\hline
[tex]C[/tex] & -20 & -15 & -5 \\
\hline
[tex]F(C)[/tex] & 23 & 5 & -4 \\
\hline
\end{tabular}

C.
\begin{tabular}{|c|c|c|c|}
\hline
[tex]C[/tex] & -20 & -15 & -5 \\
\hline
[tex]F(C)[/tex] & 4 & -5 & -23 \\
\hline
\end{tabular}

D.
\begin{tabular}{|c|c|c|c|}
\hline
[tex]C[/tex] & -20 & -15 & [tex]$\boxed{-5}$[/tex] \\
\hline
[tex]F(C)[/tex] & -23 & -5 & 4 \\
\hline
\end{tabular}



Answer :

GinnyAnswer
To determine which table could be used with Table A to verify that the function modeling Fahrenheit temperature, \(F(C)\), based on a given Celsius temperature, \(C\), is the inverse of \(C(F)\), we need to identify a table where \(F(C)\) essentially reverses the mappings provided by \(C(F)\) in Table A.

Given Table A:
[tex]\[ \begin{tabular}{|c|c|c|c|} \hline F & -4 & 5 & 23 \\ \hline C(F) & -20 & -15 & -5 \\ \hline \end{tabular} \][/tex]

We need to check which table accurately represents the inverse function \(F(C)\) such that it converts the Celsius temperatures back to the original Fahrenheit temperatures:

Option 1:
[tex]\[ \begin{tabular}{|c|c|c|c|} \hline C & -20 & -15 & -5 \\ \hline F(C) & -4 & 5 & 23 \\ \hline \end{tabular} \][/tex]

This table shows:
- \(F(-20) = -4\)
- \(F(-15) = 5\)
- \(F(-5) = 23\)

This precisely reverses the mappings given in Table A.

Option 2:
[tex]\[ \begin{tabular}{|c|c|c|c|} \hline C & -20 & -15 & -5 \\ \hline F(C) & 23 & 5 & -4 \\ \hline \end{tabular} \][/tex]

This table shows:
- \(F(-20) = 23\)
- \(F(-15) = 5\)
- \(F(-5) = -4\)

This does not reverse the mappings correctly as \(F(-20) = 23\) and \(F(-5) = -4\) contradict our initial table.

Option 3:
[tex]\[ \begin{tabular}{|c|c|c|c|} \hline C & -20 & -15 & -5 \\ \hline F(C) & 4 & -5 & -23 \\ \hline \end{tabular} \][/tex]

This table shows:
- \(F(-20) = 4\)
- \(F(-15) = -5\)
- \(F(-5) = -23\)

This also does not reverse the mappings as all the values of \(F(C)\) are incorrect.

Option 4:
[tex]\[ \begin{tabular}{|c|c|c|c|} \hline C & -20 & -15 & -5 \\ \hline F(C) & -23 & -5 & 4 \\ \hline \end{tabular} \][/tex]

This table shows:
- \(F(-20) = -23\)
- \(F(-15) = -5\)
- \(F(-5) = 4\)

This does not correctly reverse the mappings either.

Thus, the table that correctly verifies \(F(C)\) as the inverse of \(C(F)\) from Table A is the first option:

[tex]\[ \begin{tabular}{|c|c|c|c|} \hline C & -20 & -15 & -5 \\ \hline F(C) & -4 & 5 & 23 \\ \hline \end{tabular} \][/tex]

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