To find the values of [tex]\( a \)[/tex] through [tex]\( e \)[/tex] that make the given relations inverses of each other, we need to ensure that if an ordered pair [tex]\((x, y)\)[/tex] is in one relation, then the pair [tex]\((y, x)\)[/tex] is in the inverted relation. Let's examine the given tables step-by-step.
The first table indicates:
[tex]\[
\begin{array}{|r|r|}
\hline
x & y \\
\hline
-3.8 & -3.1 \\
b & 3.2 \\
-1.4 & c \\
-0.2 & 4.4 \\
1.0 & 5.0 \\
\hline
\end{array}
\][/tex]
The second table indicates:
[tex]\[
\begin{array}{|c|c|}
\hline
x & y \\
\hline
-3.1 & a \\
3.2 & -2.6 \\
1.7 & -1.4 \\
d & -0.2 \\
5.0 & e \\
\hline
\end{array}
\][/tex]
### Step-by-Step Solution:
1. Find [tex]\( a \)[/tex]:
In the first table, we have the pair [tex]\((-3.8, -3.1)\)[/tex]. In the second table, [tex]\(-3.1\)[/tex] leads to [tex]\(a\)[/tex]. Since these tables are inverses, [tex]\(-3.1 \rightarrow -3.8\)[/tex], thus:
[tex]\[
a = -3.8
\][/tex]
2. Find [tex]\( b \)[/tex]:
In the second table, we have the pair [tex]\((3.2, -2.6)\)[/tex]. The value [tex]\(3.2\)[/tex] appears in the first table as the second element of the pair [tex]\((b, 3.2)\)[/tex], which corresponds to [tex]\( -2.6 \)[/tex]. Hence:
[tex]\[
b = -2.6
\][/tex]
3. Find [tex]\( c \)[/tex]:
In the second table, we have the pair [tex]\((1.7, -1.4)\)[/tex]. From the first table, [tex]\(-1.4\)[/tex] pairs with [tex]\( c \)[/tex]. Reversing this pair should provide the value of [tex]\( c \)[/tex]:
[tex]\[
c = -1.4
\][/tex]
4. Find [tex]\( d \)[/tex]:
In the first table, we have [tex]\(( -0.2 , 4.4 )\)[/tex] and in the second table we see [tex]\(( d , -0.2 )\)[/tex]. As these tables are inverses, the pair [tex]\((-0.2, -0.2)\)[/tex] remains unchanged, so:
[tex]\[
d = -0.2
\][/tex]
5. Find [tex]\( e \)[/tex]:
From the first table, the pair [tex]\((1.0, 5.0)\)[/tex] corresponds to the second table pair [tex]\((5.0, e)\)[/tex]. Reversing the first pair [tex]\(5.0 \rightarrow 1.0\)[/tex] reveals:
[tex]\[
e = 5.0
\][/tex]
### Final Values:
[tex]\[
\begin{array}{l}
a = -3.8 \\
b = -2.6 \\
c = -1.4 \\
d = -0.2 \\
e = 5.0 \\
\end{array}
\][/tex]
Thus, the values [tex]\(a\)[/tex], [tex]\(b\)[/tex], [tex]\(c\)[/tex], [tex]\(d\)[/tex], and [tex]\(e\)[/tex] that make the given relations inverses of each other are [tex]\( -3.8, -2.6, -1.4, -0.2, \)[/tex] and [tex]\( 5.0 \)[/tex] respectively.
