To find $(f+g)(x)$ and $(f-g)(x)$ using the given functions $f(x)$ and $g(x)$, we perform the following operations:

1. Given Functions:

• $f(x) = 3x + 3$
• $g(x) = -4x + 1$

$(f+g)(x)$:

$(f+g)(x) = f(x) + g(x)$ Substitute the expressions for $f(x)$ and $g(x)$: $(f+g)(x) = (3x + 3) + (-4x + 1)$ Simplify: $(f+g)(x) = 3x - 4x + 3 + 1 = -x + 4$

$(f-g)(x)$:

$(f-g)(x) = f(x) - g(x)$ Substitute the expressions for $f(x)$ and $g(x)$: $(f-g)(x) = (3x + 3) - (-4x + 1)$ Simplify: $(f-g)(x) = 3x + 3 + 4x - 1 = 7x + 2$

Final Answers:

• $(f+g)(x) = -x + 4$
• $(f-g)(x) = 7x + 2$

Would you like more details or have any questions?

Here are some related questions:

1. How do you find $(f \cdot g)(x)$ using the given functions?
2. What is $(f/g)(x)$ for these functions?
3. How would $(f+g)(x)$ change if $g(x)$ was a quadratic function?
4. What is the value of $(f+g)(2)$?
5. How would the process differ if $f(x)$ and $g(x)$ were trigonometric functions?
6. What is the graphical interpretation of $(f+g)(x)$?
7. Can you express $f(x)$ and $g(x)$ as a single combined function?
8. What is the domain of the function $(f-g)(x)$?

Tip: When combining functions, always simplify the expression to its most reduced form for clarity.