Given the functions $f(x) = 4x + 5$ and $g(x) = 3x + 4$, you can perform several operations with them. Here are some common ones:

1. Sum of the functions: $(f + g)(x) = f(x) + g(x)$ Substitute the given functions: $(f + g)(x) = (4x + 5) + (3x + 4) = 7x + 9$

2. Difference of the functions: $(f - g)(x) = f(x) - g(x)$ Substitute the given functions: $(f - g)(x) = (4x + 5) - (3x + 4) = x + 1$

3. Product of the functions: $(f \cdot g)(x) = f(x) \cdot g(x)$ Substitute the given functions: $(f \cdot g)(x) = (4x + 5) \cdot (3x + 4)$ Expanding the product: $(f \cdot g)(x) = 12x^2 + 16x + 15x + 20 = 12x^2 + 31x + 20$

4. Quotient of the functions: $\left(\frac{f}{g}\right)(x) = \frac{f(x)}{g(x)}$ Substitute the given functions: $\left(\frac{f}{g}\right)(x) = \frac{4x + 5}{3x + 4}$

5. Composition of the functions (two cases):

   • $f(g(x))$: $f(g(x)) = f(3x + 4)$ Substitute $g(x)$ into $f(x)$: $f(g(x)) = 4(3x + 4) + 5 = 12x + 16 + 5 = 12x + 21$
   • $g(f(x))$: $g(f(x)) = g(4x + 5)$ Substitute $f(x)$ into $g(x)$: $g(f(x)) = 3(4x + 5) + 4 = 12x + 15 + 4 = 12x + 19$

Would you like more details or have any questions?

Here are five related questions:

1. What is the derivative of the function $f(x) = 4x + 5$?
2. How do you find the intersection points of the graphs of $f(x)$ and $g(x)$?
3. What is the inverse of $f(x) = 4x + 5$?
4. Can you solve for $x$ when $f(x) = g(x)$?
5. How would you graph $f(x)$ and $g(x)$ on the same coordinate plane?

Tip: Always check the domain of your functions, especially when dealing with compositions or quotients, to avoid undefined values.