Question 1165124
Here are the solutions for the inverse function problems.

## 1. Find $g^{-1}(1)$

The function $g$ is defined by a set of ordered pairs:
$$g = \{(-7, 1), (-4, 5), (1, -1), (7, 3)\}$$

The inverse function, $g^{-1}$, reverses the ordered pairs: if $(a, b)$ is in $g$, then $(b, a)$ is in $g^{-1}$.
$$g^{-1} = \{(1, -7), (5, -4), (-1, 1), (3, 7)\}$$

To find $g^{-1}(1)$, look for the pair in $g^{-1}$ where the input (x-value) is 1.
The pair is $(1, -7)$.

$$\mathbf{g^{-1}(1) = -7}$$

*Alternatively, we look for the output (y-value) of 1 in the original function $g$. Since $g(-7) = 1$, then $g^{-1}(1) = -7$.*

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

The function $h(x)$ is given by:
$$h(x) = 4x - 9$$

To find the inverse function, $h^{-1}(x)$:
1.  **Replace $h(x)$ with $y$**:
    $$y = 4x - 9$$
2.  **Swap $x$ and $y$**:
    $$x = 4y - 9$$
3.  **Solve for $y$**:
    $$x + 9 = 4y$$
    $$y = \frac{x + 9}{4}$$
4.  **Replace $y$ with $h^{-1}(x)$**:

$$\mathbf{h^{-1}(x) = \frac{x + 9}{4}}$$

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

The composition $(h^{-1} \circ h)(x)$ is defined as $h^{-1}(h(x))$.

Since $h(x)$ and $h^{-1}(x)$ are inverses of each other, their composition always returns the original input, $x$, for any value in the domain.

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

Therefore, for the input $x=1$:
$$(h^{-1} \circ h)(1) = 1$$

*Alternatively, solving step-by-step:*
1.  **Find $h(1)$**:
    $$h(1) = 4(1) - 9 = 4 - 9 = -5$$
2.  **Find $h^{-1}(h(1))$, which is $h^{-1}(-5)$**:
    $$h^{-1}(-5) = \frac{(-5) + 9}{4} = \frac{4}{4} = 1$$

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