The image contains multiple problems involving functions $f(x)$ and $g(x)$. The tasks seem to focus on composing functions (such as $f(g(x))$ and $g(f(x))$) and evaluating the compositions at specific values. Below is a detailed outline of the problems from the image, along with explanations for each.

Problems Summary:

1. $f(x) = x - 1$, $g(x) = 2x - 1$

• Find $f(g(x))$ and $g(f(x))$.

2. $f(x) = 2 + 3x$, $g(x) = x - 3$

• Find $f(g(x))$ and $g(f(x))$.

3. $f(x) = 2x - 5$, $g(x) = x - 3x$

• Find $f(g(x))$ and $g(f(x))$.

4. $f(x) = 3x - 2$, $g(x) = \frac{3 - 2x}{2x - 1}$

• Find $f(g(x))$ and $g(f(x))$.

5. $f(x) = x + 1$, $g(x) = \frac{2x+1}{3 - 2x}$

• Tasks:
  • (a) Find $(f \circ g)(x)$ (i.e., $f(g(x))$).
  • (b) Find $(g \circ f)(x)$ (i.e., $g(f(x))$).
  • (c) Evaluate $(f \circ g)(1)$.
  • (d) Evaluate $(g \circ f)(-1)$.

6. $f(x) = 2 - 5x$, $g(x) = \frac{x+1}{2x-1}$

• Find $f(g(x))$ and $g(f(x))$.

7. $f(x) = \frac{3x+2}{3-x}$, $g(x) = 3 - 2x$

• Find $f(g(x))$ and $g(f(x))$.

8. $f(x) = x^2 + 2x + 1$, $g(x) = 3x - 1$

• Find $f(g(x))$ and $g(f(x))$.

9. $f(x) = 2 - 3x$, $g(x) = 2 - x - x^2$

• Find $f(g(x))$ and $g(f(x))$.

10. $f(x) = \frac{2x + 1}{x + 2}$, $g(x) = 2x + 3$

• Find $f(g(x))$ and $g(f(x))$.

Would you like me to begin solving each of these problems one by one, or would you prefer specific ones to be solved first? Let me know if you want detailed explanations for each solution!

Follow-up Questions:

1. How is function composition different from addition or multiplication of functions?
2. What happens when you compose two inverse functions?
3. How do you evaluate the composition of more complex rational functions?
4. How does changing the order of composition affect the result?
5. What are some real-world applications of function composition in different fields?

Tip: When composing functions, always be careful with the domain restrictions, especially for rational functions where denominators can lead to undefined values.