Let's break down the solution step by step:
### 1. Finding [tex]\( g^{-1}(6) \)[/tex]
#### Given:
[tex]\[ g = \{(-3, -2), (2, -9), (3, 6), (6, 5)\} \][/tex]
We need to determine [tex]\( g^{-1}(6) \)[/tex], which means finding [tex]\( x \)[/tex] such that [tex]\( g(x) = 6 \)[/tex].
To do this, we will look for the pair [tex]\((x, y)\)[/tex] in the set [tex]\( g \)[/tex] where [tex]\( y = 6 \)[/tex]. By examining the pairs:
- [tex]\( (-3, -2) \)[/tex]
- [tex]\( (2, -9) \)[/tex]
- [tex]\( (3, 6) \)[/tex]
- [tex]\( (6, 5) \)[/tex]
We find that [tex]\( g(3) = 6 \)[/tex].
Thus, [tex]\( g^{-1}(6) = 3 \)[/tex].
### 2. Finding [tex]\( h^{-1}(x) \)[/tex]
#### Given:
[tex]\[ h(x) = 2x - 9 \][/tex]
We need to find [tex]\( h^{-1}(x) \)[/tex], the inverse function of [tex]\( h(x) \)[/tex]. To do this, we solve for [tex]\( x \)[/tex] in terms of [tex]\( y \)[/tex] from the equation [tex]\( y = 2x - 9 \)[/tex].
1. Start with the equation [tex]\( y = 2x - 9 \)[/tex].
2. Add 9 to both sides:
[tex]\[ y + 9 = 2x \][/tex]
3. Divide both sides by 2:
[tex]\[ x = \frac{y + 9}{2} \][/tex]
Thus, the inverse function [tex]\( h^{-1}(x) \)[/tex] is:
[tex]\[ h^{-1}(x) = \frac{x + 9}{2} \][/tex]
### 3. Finding [tex]\( (h^{-1} \circ h)(7) \)[/tex]
To find [tex]\( (h^{-1} \circ h)(7) \)[/tex], we first compute [tex]\( h(7) \)[/tex], and then apply [tex]\( h^{-1} \)[/tex] to the result.
#### Step 1: Compute [tex]\( h(7) \)[/tex]
Given [tex]\( h(x) = 2x - 9 \)[/tex]:
[tex]\[ h(7) = 2 \cdot 7 - 9 = 14 - 9 = 5 \][/tex]
#### Step 2: Apply [tex]\( h^{-1} \)[/tex] to [tex]\( h(7) \)[/tex]
We now apply [tex]\( h^{-1} \)[/tex] to the result found:
[tex]\[ h^{-1}(5) = \frac{5 + 9}{2} = \frac{14}{2} = 7 \][/tex]
Thus, [tex]\( (h^{-1} \circ h)(7) = 7 \)[/tex].
### Summary of Results
- [tex]\( g^{-1}(6) = 3 \)[/tex]
- [tex]\( h^{-1}(x) = \frac{x + 9}{2} \)[/tex]
- [tex]\( (h^{-1} \circ h)(7) = 7 \)[/tex]
These detailed steps provide a clear mathematical solution to the given question.
