Certainly! Let's explore the functions [tex]\(g\)[/tex] and [tex]\(h(x)\)[/tex] defined as follows:
[tex]\[
\begin{array}{l}
g = \{(-9, -3), (2, 8), (3, -5), (8, -9)\} \\
h(x) = 3x - 2
\end{array}
\][/tex]
### Step 1: List the results of [tex]\( g(x) \)[/tex]:
Let's map each input [tex]\( x_i \)[/tex] to their corresponding outputs using [tex]\( g \)[/tex]:
[tex]\[
\begin{array}{lcl}
g(-9) &=& -3, \\
g(2) &=& 8, \\
g(3) &=& -5, \\
g(8) &=& -9.
\end{array}
\][/tex]
So, we have the results of [tex]\( g(x_i) \)[/tex] as:
[tex]\[
[(-9, -3), (2, 8), (3, -5), (8, -9)]
\][/tex]
### Step 2: Calculate the results of [tex]\( h(x) \)[/tex]:
Now we need to find the value of [tex]\( h(x) \)[/tex] for each output value from [tex]\( g \)[/tex]:
[tex]\[
\begin{array}{lcl}
h(-3) &=& 3(-3) - 2 = -9 - 2 = -11, \\
h(8) &=& 3(8) - 2 = 24 - 2 = 22, \\
h(-5) &=& 3(-5) - 2 = -15 - 2 = -17, \\
h(-9) &=& 3(-9) - 2 = -27 - 2 = -29.
\end{array}
\][/tex]
Thus, we have the results of [tex]\( h(x_i) \)[/tex] as:
[tex]\[
[(-3, -11), (8, 22), (-5, -17), (-9, -29)]
\][/tex]
### Step 3: Find the composition [tex]\( g(h(x)) \)[/tex]:
For the composition [tex]\( g(h(x)) \)[/tex], we need to evaluate [tex]\( g(h(x)) \)[/tex] for each input [tex]\( x_i \)[/tex] in the domain of [tex]\( g \)[/tex]:
To find [tex]\( g(h(x_i)) \)[/tex]:
1. We assume inputs [tex]\( x_i \)[/tex] that are the keys of [tex]\( g \)[/tex].
2. We first compute [tex]\( h(x_i) \)[/tex] and then check if the result is in the domain of [tex]\( g \)[/tex]:
[tex]\[
\begin{array}{lcl}
h(-9) &=& -29, \quad \text{but } -29 \text{ is not in the domain of } g, \\
h(2) &=& 4, \quad \text{but } 4 \text{ is not in the domain of } g, \\
h(3) &=& 7, \quad \text{but } 7 \text{ is not in the domain of } g, \\
h(8) &=& 22, \quad \text{but } 22 \text{ is not in the domain of } g.
\end{array}
\][/tex]
As none of the results from [tex]\( h(x_i) \)[/tex] using keys from [tex]\( g \)[/tex] are found in the domain of [tex]\( g \)[/tex], the composition [tex]\( g(h(x_i)) \)[/tex] is empty.
Therefore, the composition [tex]\( g(h(x)) \)[/tex] results in:
[tex]\[
[]
\][/tex]
### Summary:
1. Results of [tex]\( g(x_i) \)[/tex]:
[tex]\[
[(-9, -3), (2, 8), (3, -5), (8, -9)]
\][/tex]
2. Results of [tex]\( h(x_i) \)[/tex]:
[tex]\[
[(-3, -11), (8, 22), (-5, -17), (-9, -29)]
\][/tex]
3. Composition [tex]\( g(h(x_i)) \)[/tex]:
[tex]\[
[]
\][/tex]
This provides a complete solution to the problem posed.
