Find the inverse functions and compositions of the given functions f(x) = 16x + 12, g(x) = (3x + 2)/(6), and h(x) = 2x − 2.

$$x - 12 = 16y$$ $$y = \frac{x - 12}{16}$$

$$\boxed{f^{-1}(x) = \frac{x - 12}{16}}$$

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

$$6x = 3y + 2$$ $$6x - 2 = 3y$$ $$y = \frac{6x - 2}{3}$$

$$\boxed{g^{-1}(x) = \frac{6x - 2}{3}}$$

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

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

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

4. Find $f \circ g$, $g \circ h$, $h \circ g$

• Find $f \circ g(x)$

  $$f \circ g(x) = f(g(x))$$ $$f(g(x)) = 16\left(\frac{3x + 2}{6}\right) + 12$$ $$f(g(x)) = \frac{16(3x + 2)}{6} + 12$$ $$f(g(x)) = \frac{8(3x + 2)}{3} + 12$$ $$f(g(x)) = \frac{24x + 16}{3} + \frac{36}{3}$$ $$f(g(x)) = \frac{24x + 16 + 36}{3}$$ $$\boxed{f \circ g(x) = \frac{24x + 52}{3}}$$

• Find $g \circ h(x)$

  $$g \circ h(x) = g(h(x))$$ $$g(h(x)) = \frac{3(2x - 2) + 2}{6}$$ $$g(h(x)) = \frac{6x - 6 + 2}{6}$$ $$g(h(x)) = \frac{6x - 4}{6}$$ $$g(h(x)) = \frac{2(3x - 2)}{6}$$ $$\boxed{g \circ h(x) = \frac{3x - 2}{3}}$$

• Find $h \circ g(x)$

  $$h \circ g(x) = h(g(x))$$ $$h(g(x)) = 2\left(\frac{3x + 2}{6}\right) - 2$$ $$h(g(x)) = \frac{2(3x + 2)}{6} - 2$$ $$h(g(x)) = \frac{3x + 2}{3} - 2$$ $$h(g(x)) = \frac{3x + 2}{3} - \frac{6}{3}$$ $$h(g(x)) = \frac{3x + 2 - 6}{3}$$ $$\boxed{h \circ g(x) = \frac{3x - 4}{3}}$$

5. Find $f^{-1}(g(x))$ and $h^{-1}(g(x))$

• Find $f^{-1}(g(x))$ Substitute $g(x) = \frac{3x + 2}{6}$ into $f^{-1}(x) = \frac{x - 12}{16}$.

  $$f^{-1}(g(x)) = \frac{\left(\frac{3x + 2}{6}\right) - 12}{16}$$ $$f^{-1}(g(x)) = \frac{\frac{3x + 2}{6} - \frac{72}{6}}{16}$$ $$f^{-1}(g(x)) = \frac{\frac{3x + 2 - 72}{6}}{16}$$ $$f^{-1}(g(x)) = \frac{3x - 70}{6 \times 16}$$ $$\boxed{f^{-1}(g(x)) = \frac{3x - 70}{96}}$$

• Find $h^{-1}(g(x))$ Substitute $g(x) = \frac{3x + 2}{6}$ into $h^{-1}(x) = \frac{x + 2}{2}$.

  $$h^{-1}(g(x)) = \frac{\left(\frac{3x + 2}{6}\right) + 2}{2}$$ $$h^{-1}(g(x)) = \frac{\frac{3x + 2}{6} + \frac{12}{6}}{2}$$ $$h^{-1}(g(x)) = \frac{\frac{3x + 2 + 12}{6}}{2}$$ $$h^{-1}(g(x)) = \frac{3x + 14}{6 \times 2}$$ $$\boxed{h^{-1}(g(x)) = \frac{3x + 14}{12}}$$