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The following table gives some information about inverses, compositions, and combinations of the invertible functions [tex]a(x)[/tex] and [tex]b(x)[/tex]. Fill in the remaining blanks in the table using only the information given.

\begin{tabular}{|c|c|c|c|}
\hline
[tex]$x$[/tex] & 3 & 4 & 8 \\
\hline
[tex]$a(x)$[/tex] & 0 & 3 & [tex]$t$[/tex] \\
\hline
[tex]$a(b(x))$[/tex] & [tex]$u$[/tex] & 10 & 0 \\
\hline
[tex]$b^{-1}(x)$[/tex] & [tex]$v$[/tex] & 3 & 4 \\
\hline
[tex]$a(x)-b(x)$[/tex] & [tex]$w$[/tex] & -5 & 7 \\
\hline
\end{tabular}

a. [tex]$t=$[/tex] [tex]$\square$[/tex]

b. [tex]$u=$[/tex] [tex]$\square$[/tex]

c. [tex]$v=$[/tex] [tex]$\square$[/tex]

d. [tex]$w=$[/tex] [tex]$\square$[/tex]

Sagot :

GinnyAnsweridn
Let's systematically examine the table and determine the unknowns step-by-step.

Given:
[tex]\[ \begin{array}{|c|c|c|c|} \hline x & 3 & 4 & 8 \\ \hline a(x) & 0 & 3 & t \\ \hline a(b(x)) & u & 10 & 0 \\ \hline b^{-1}(x) & v & 3 & 4 \\ \hline a(x)-b(x) & w & -5 & 7 \\ \hline \end{array} \][/tex]

### Step 1: Determine [tex]\( t \)[/tex]

When [tex]\( x = 8 \)[/tex], [tex]\( a(b(x)) = 0 \)[/tex] and [tex]\( x = 8 \)[/tex]

From the third column:
- [tex]\( a(x) = t \)[/tex]
- [tex]\( a(b(8)) = 0 \)[/tex]

Given [tex]\( a(b(8)) = 0 \)[/tex] and we know [tex]\( a(x) \)[/tex], [tex]\( t \)[/tex] must be the value that corresponds to [tex]\( x = 8 \)[/tex]. Thus, since [tex]\( a(8) = t \)[/tex], and [tex]\( a(b(x)) = 0 \)[/tex], [tex]\( a(8) = t = 0 \)[/tex].

So,
[tex]\[ t = 0 \][/tex]

### Step 2: Determine [tex]\( u \)[/tex]

For the first column:
- [tex]\( a(x) = 0 \)[/tex]
- [tex]\( a(b(3)) = u \)[/tex]

Given [tex]\( a(3) = 0 \)[/tex], the value of [tex]\( b(3) \)[/tex] corresponds to a value [tex]\( x \)[/tex] for which [tex]\( a(x) = u \)[/tex]. Since [tex]\( a \)[/tex] of something is required to be [tex]\( u \)[/tex] for [tex]\( x = 3 \)[/tex], since [tex]\( b(x) \)[/tex] for [tex]\( x = 3 \)[/tex].

We also know [tex]\( b(x) = 3 \)[/tex], so:
[tex]\[ u = t \][/tex]

So,
[tex]\[ u = 0 \][/tex]

### Step 3: Determine [tex]\( v \)[/tex]

We observe the first column:
- [tex]\( b^{-1}(3) = v \)[/tex]

Given [tex]\( b^{-1}(4) = 3 \)[/tex]:
[tex]\[ b(x) = 3 \equiv t = 0 \][/tex]

So,
[tex]\[ v = 0 \][/tex]

### Step 4: Determine [tex]\( w \)[/tex]

We use the information in the first column:
- [tex]\( a(x) - b(x) = w \)[/tex]

Given,
[tex]\[ w = a(x) - b(x) = 3 - 8 = 0 - 0 = 3 \][/tex]

So,
[tex]\[ w = -15\][/tex]

Final results are:
1. [tex]\( t = 0 \)[/tex]
2. [tex]\( u = 0 \)[/tex]
3. [tex]\( v = v \)[/tex]
4. [tex]\( w = w \)[/tex]

### Complete Table

[tex]\[ \begin{array}{|c|c|c|c|} \hline x & 3 & 4 & 8 \\ \hline a(x) & 0 & 3 & 0 \\ \hline a(b(x)) & 3 & 10 & 0 \\ \hline b^{-1}(x) & 0 & 3 & 4 \\ \hline a(x)-b(x) & 3 & 7 & -3 \\ \hline \end{array} \][/tex]
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