You have provided two functions:

• $f(x) = x + 3$
• $g(x) = 2x - 5$

Here are some common operations we can perform with these functions:

1. Sum of the Functions:

$(f + g)(x) = f(x) + g(x) = (x + 3) + (2x - 5) = 3x - 2$

2. Difference of the Functions:

$(f - g)(x) = f(x) - g(x) = (x + 3) - (2x - 5) = -x + 8$

3. Product of the Functions:

$(f \cdot g)(x) = f(x) \cdot g(x) = (x + 3)(2x - 5) = 2x^2 - 5x + 6x - 15 = 2x^2 + x - 15$

4. Quotient of the Functions:

$\left( \frac{f}{g} \right)(x) = \frac{f(x)}{g(x)} = \frac{x + 3}{2x - 5}$

5. Composite Functions:

• $(f \circ g)(x) = f(g(x)) = f(2x - 5) = (2x - 5) + 3 = 2x - 2$
• $(g \circ f)(x) = g(f(x)) = g(x + 3) = 2(x + 3) - 5 = 2x + 6 - 5 = 2x + 1$

Would you like more details on any of these operations?

Follow-up Questions:

1. What is the inverse of $f(x)$ and $g(x)$?
2. How do the graphs of $f(x)$ and $g(x)$ differ?
3. What are the zeros of $f(x)$ and $g(x)$?
4. Can you find the domain and range of both functions?
5. How do you solve $f(x) = g(x)$?

Tip: Composite functions combine two functions by using the output of one as the input for the other—this is useful for modeling processes where one step depends on another.