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.

Given the functions f(x)=4x+5 and g(x)=3x+4, perform operations like sum, difference, product, quotient, and composition of functions.

Explore how to perform operations on functions f(x) = 4x + 5 and g(x) = 3x + 4, including sum, difference, product, quotient, and composition. This guide is suitable for high school students in grades 9-12, covering key algebraic concepts and formulas.

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