Question 1165630
This problem involves finding inverse functions and evaluating a function composition.

---

## 1. Finding $g^{-1}(8)$

The function $g$ is defined by the set of ordered pairs:
$$g = \{(-4,-9), (-2,8), (4,5), (8,4)\}$$

The inverse function $g^{-1}$ reverses the ordered pairs. To find $g^{-1}(8)$, we look for the pair in $g$ where the output (y-value) is 8.

The pair in $g$ with an output of 8 is **$(-2, 8)$**.

Therefore, the input (x-value) for $g^{-1}(8)$ is $-2$.

$$\mathbf{g^{-1}(8) = -2}$$

---

## 2. Finding $h^{-1}(x)$

The function is $h(x) = 2x - 13$. To find the inverse, we set $y = h(x)$ and solve for $x$ in terms of $y$.

1.  Set $y = h(x)$:
    $$y = 2x - 13$$
2.  Swap $x$ and $y$:
    $$x = 2y - 13$$
3.  Solve for $y$:
    $$x + 13 = 2y$$
    $$y = \frac{x + 13}{2}$$

$$\mathbf{h^{-1}(x) = \frac{x + 13}{2}}$$

---

## 3. Finding $(h \circ h^{-1})(-5)$

The expression $(h \circ h^{-1})(-5)$ represents the composition of the function $h$ with its inverse $h^{-1}$, evaluated at $x=-5$.

By the definition of an inverse function, when a function is composed with its inverse, the result is the identity function:
$$(f \circ f^{-1})(x) = x$$

Therefore, for $h(x)$:

$$(h \circ h^{-1})(-5) = -5$$

$$\mathbf{(h \circ h^{-1})(-5) = -5}$$