To verify whether the functions represented by Table A and Table B are inverses of each other, we need to check the correspondence between the pairs in the tables.
Table A
[tex]\[
\begin{array}{|c|c|c|c|}
\hline
x & 0.75 & 1 & 1.25 \\
\hline
a(x) & 1038.18 & 1051.21 & 1064.39 \\
\hline
\end{array}
\][/tex]
Table B
[tex]\[
\begin{array}{|c|c|c|c|}
\hline
d & 1057.81 & 1077.78 & 1098.12 \\
\hline
r(d) & 0.75 & 1 & 1.25 \\
\hline
\end{array}
\][/tex]
To determine if the functions are inverses, we need to verify if for each [tex]\((x, a(x))\)[/tex] pair in Table A, there is a corresponding [tex]\((a(x), x)\)[/tex] pair in Table B. This means:
- For [tex]\(x = 0.75\)[/tex] in Table A:
- [tex]\(a(x) = 1038.18\)[/tex]
- Check if [tex]\(d = 1038.18\)[/tex] in Table B and [tex]\(r(d) = 0.75\)[/tex].
- In Table B, [tex]\(d = 1038.18\)[/tex] is not present.
- For [tex]\(x = 1\)[/tex] in Table A:
- [tex]\(a(x) = 1051.21\)[/tex]
- Check if [tex]\(d = 1051.21\)[/tex] in Table B and [tex]\(r(d) = 1\)[/tex].
- In Table B, [tex]\(d = 1051.21\)[/tex] is not present.
- For [tex]\(x = 1.25\)[/tex] in Table A:
- [tex]\(a(x) = 1064.39\)[/tex]
- Check if [tex]\(d = 1064.39\)[/tex] in Table B and [tex]\(r(d) = 1.25\)[/tex].
- In Table B, [tex]\(d = 1064.39\)[/tex] is not present.
Since there are no corresponding [tex]\( (d, r(d)) \)[/tex] pairs in Table B for the [tex]\((x, a(x))\)[/tex] pairs in Table A, we conclude that the functions are not inverses.
Thus, the correct conclusion is:
"The functions are not inverses because for each ordered pair [tex]\((x, y)\)[/tex] for one function, there is no corresponding ordered pair [tex]\((y, x)\)[/tex] for the other function."
Hence, the answer would be:
"The functions are not inverses because for each ordered pair [tex]\((x, y)\)[/tex] for one function, there is no corresponding ordered pair [tex]\((y, x)\)[/tex] for the other function."
