The image you uploaded contains a set of mathematical exercises related to function composition and transformation. I'll transcribe the content for clarity and help solve any or all of the problems listed.

1. Let $f(x) = 2x + 5$ and $g(x) = 3x - 4$. Build a new function $h(x)$ for which $h(x) = f(g(x))$.

2. Let $f(x) = 4x - 1$ and $g(x) = \frac{1}{2}x + 3$. Build a new function $h(x)$ for which $h(x) = g(f(x))$.

3. Let $f(x) = 2x + 3$ and let $h(x)$ be a horizontal transformation, such that $h(x) = f(2x)$.

4. Let $f(x) = x^2 - 2$, find $g(x) = \frac{x^2}{2} + 1$. Build a new function $h(x)$, such that $h(x) = \frac{g(x)}{f(x)}$. State any restrictions on the domain and range.

5. Let $g(x) = 5x + 7$ and $f(x) = 6x - 2$. Build a new function $h(x)$, for which $h(x) = f(g(x))$.

Would you like me to solve one or more of these problems? Let me know which ones you want to focus on.

Here are some related questions that might deepen your understanding:

1. How do function compositions work in terms of domain restrictions?
2. What are common transformations like horizontal stretching or shrinking?
3. Can you explain how to find the inverse of a composite function?
4. What are the impacts of applying transformations to functions graphically?
5. How would you solve for $x$ in a composed function $h(x) = f(g(x))$?

Tip: When working with composed functions, always ensure you evaluate the domain restrictions from the inside function first.