We are given two problems. The first problem asks us to find the inverse function $h^{-1}(y)$ of the function $h(x) = \frac{3}{2}x - 7$. The second problem asks us to find the inverse function $f^{-1}(y)$ of the function $f(x) = \sqrt{x-2} + 4$.

1. Problem Description

We are given two problems. The first problem asks us to find the inverse function $h^{-1}(y)$ of the function $h(x) = \frac{3}{2}x - 7$. The second problem asks us to find the inverse function $f^{-1}(y)$ of the function $f(x) = \sqrt{x-2} + 4$.

2. Solution Steps

Problem 1:

We are given $h(x) = \frac{3}{2}x - 7$. To find the inverse function $h^{-1}(y)$, we set $y = h(x)$ and solve for $x$ in terms of $y$.

$y = \frac{3}{2}x - 7$

$y + 7 = \frac{3}{2}x$

$x = \frac{2}{3}(y + 7)$

$x = \frac{2y + 14}{3}$

Thus, $h^{-1}(y) = \frac{2y + 14}{3}$.

Problem 2:

We are given $f(x) = \sqrt{x-2} + 4$. To find the inverse function $f^{-1}(y)$, we set $y = f(x)$ and solve for $x$ in terms of $y$.

$y = \sqrt{x-2} + 4$

$y - 4 = \sqrt{x-2}$

$(y - 4)^2 = x - 2$

$x = (y - 4)^2 + 2$

Thus, $f^{-1}(y) = (y - 4)^2 + 2$.

3. Final Answer

$h^{-1}(y) = \frac{2y + 14}{3}$

$f^{-1}(y) = (y - 4)^2 + 2$