To find $(f+g)(x)$ and $(f-g)(x)$ using the functions $f(x) = 3x + 3$ and $g(x) = -4x + 1$, follow these steps:

1. Find $(f+g)(x)$:

$(f+g)(x) = f(x) + g(x)$ Substitute the given functions: $(f+g)(x) = (3x + 3) + (-4x + 1)$ Combine like terms: $(f+g)(x) = 3x - 4x + 3 + 1 = -x + 4$ So, $(f+g)(x) = -x + 4$

2. Find $(f-g)(x)$:

$(f-g)(x) = f(x) - g(x)$ Substitute the given functions: $(f-g)(x) = (3x + 3) - (-4x + 1)$ Distribute the negative sign: $(f-g)(x) = 3x + 3 + 4x - 1$ Combine like terms: $(f-g)(x) = 3x + 4x + 3 - 1 = 7x + 2$ So, $(f-g)(x) = 7x + 2$

Summary:

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

Would you like more details or have any questions?

Here are five related questions to further explore this topic:

1. How would you find $(f \times g)(x)$ using the same functions?
2. What is the graph of $(f+g)(x)$ and $(f-g)(x)$?
3. How do you find the domain and range of $(f+g)(x)$ and $(f-g)(x)$?
4. What happens if you divide $f(x)$ by $g(x)$ and find $\frac{f}{g}(x)$?
5. Can you solve for the values of $x$ where $(f+g)(x) = 0$ and $(f-g)(x) = 0$?

Tip: When combining functions, always simplify the expression by combining like terms to get the final result.