Let's solve the problem step by step:
1. Finding [tex]\( g^{-1}(-7) \)[/tex]:
- First, we need to understand that [tex]\( g^{-1} \)[/tex] is the inverse of the function [tex]\( g \)[/tex]. This means we swap the input and output pairs of [tex]\( g \)[/tex].
- Given [tex]\( g = \{(-7, 8), (-1, -7), (7, -6), (8, 4)\} \)[/tex], the pairs are:
[tex]\[
g(-7) = 8, \quad g(-1) = -7, \quad g(7) = -6, \quad g(8) = 4
\][/tex]
- For the inverse function [tex]\( g^{-1} \)[/tex], the pairs will be:
[tex]\[
g^{-1}(8) = -7, \quad g^{-1}(-7) = -1, \quad g^{-1}(-6) = 7, \quad g^{-1}(4) = 8
\][/tex]
- Therefore, [tex]\( g^{-1}(-7) = -1 \)[/tex].
2. Finding [tex]\( h^{-1}(x) \)[/tex]:
- To find the inverse of [tex]\( h(x) = 3x - 4 \)[/tex], we need to solve for [tex]\( x \)[/tex] in terms of [tex]\( y \)[/tex], where [tex]\( y = h(x) \)[/tex].
- Starting with [tex]\( y = 3x - 4 \)[/tex], we solve for [tex]\( x \)[/tex]:
[tex]\[
y = 3x - 4 \quad \Rightarrow \quad y + 4 = 3x \quad \Rightarrow \quad x = \frac{y + 4}{3}
\][/tex]
- Thus, the inverse function [tex]\( h^{-1}(x) \)[/tex] is given by [tex]\( h^{-1}(x) = \frac{x + 4}{3} \)[/tex].
3. Finding [tex]\( \left(h \circ h^{-1}\right)(2) \)[/tex]:
- To solve [tex]\( \left(h \circ h^{-1}\right)(2) \)[/tex], we need to evaluate [tex]\( h(h^{-1}(2)) \)[/tex].
- First, we find [tex]\( h^{-1}(2) \)[/tex]:
[tex]\[
h^{-1}(2) = \frac{2 + 4}{3} = \frac{6}{3} = 2
\][/tex]
- Next, substitute this value back into [tex]\( h(x) \)[/tex]:
[tex]\[
h(2) = 3 \times 2 - 4 = 6 - 4 = 2
\][/tex]
- Therefore, [tex]\( \left(h \circ h^{-1}\right)(2) = 2 \)[/tex].
In conclusion:
[tex]\[
\begin{array}{|r|}
\hline g^{-1}(-7) = -1 \\
h^{-1}(x) = \frac{x + 4}{3} \\
\left(h \circ h^{-1}\right)(2) = 2 \\
\end{array}
\][/tex]
