Let's solve the problem step by step.
1. Find [tex]\( g^{-1}(4) \)[/tex]:
- The function [tex]\( g \)[/tex] is given by the set of ordered pairs [tex]\( \{(0,-2), (1,7), (4,-9), (9,4)\} \)[/tex].
- To find the inverse function value [tex]\( g^{-1}(4) \)[/tex], we look for the pair where [tex]\( 4 \)[/tex] is the input (first element) in the function [tex]\( g \)[/tex].
- Checking the pairs in [tex]\( g \)[/tex], we find that [tex]\( g(4) = -9 \)[/tex].
- Thus, the inverse function [tex]\( g^{-1} \)[/tex] will give us the value [tex]\( g^{-1}(4) = -9 \)[/tex].
2. Find the inverse function [tex]\( h^{-1}(x) \)[/tex]:
- The function [tex]\( h(x) \)[/tex] is given by [tex]\( h(x) = 3x - 10 \)[/tex].
- To find the inverse [tex]\( h^{-1}(x) \)[/tex], we need to solve for [tex]\( x \)[/tex] in terms of [tex]\( y \)[/tex].
- Start with the equation [tex]\( y = 3x - 10 \)[/tex].
- Add 10 to both sides: [tex]\( y + 10 = 3x \)[/tex].
- Now, divide both sides by 3: [tex]\( x = \frac{y + 10}{3} \)[/tex].
- Therefore, the inverse function [tex]\( h^{-1}(x) \)[/tex] is [tex]\( h^{-1}(x) = \frac{x + 10}{3} \)[/tex].
3. Evaluate [tex]\( \left(h^{-1} \circ h\right)(5) \)[/tex]:
- The composition [tex]\( (h^{-1} \circ h)(x) \)[/tex] means first applying [tex]\( h \)[/tex] to [tex]\( x \)[/tex], and then applying [tex]\( h^{-1} \)[/tex] to the result.
- Let's find [tex]\( h(5) \)[/tex]:
[tex]\[
h(5) = 3 \cdot 5 - 10 = 15 - 10 = 5
\][/tex]
- Now apply [tex]\( h^{-1} \)[/tex] to [tex]\( h(5) \)[/tex]:
[tex]\[
h^{-1}(5) = \frac{5 + 10}{3} = \frac{15}{3} = 5
\][/tex]
- Thus, [tex]\( \left(h^{-1} \circ h\right)(5) = 5 \)[/tex].
Summarizing the results:
[tex]\[
\begin{array}{|c|}
\hline
g^{-1}(4) = -9 \\
h^{-1}(x) = \frac{x + 10}{3} \\
\left(h^{-1} \circ h\right)(5) = 5 \\
\hline
\end{array}
\][/tex]
