To find $(f+g)(x)$ and $(f-g)(x)$ using the functions $f(x) = 5x + 1$ and $g(x) = -2 + 3x$, follow these steps:

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

$(f+g)(x) = f(x) + g(x)$ Substitute the given functions: $(f+g)(x) = (5x + 1) + (-2 + 3x)$ Combine like terms: $(f+g)(x) = 5x + 3x + 1 - 2 = 8x - 1$ So, $(f+g)(x) = 8x - 1$

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

$(f-g)(x) = f(x) - g(x)$ Substitute the given functions: $(f-g)(x) = (5x + 1) - (-2 + 3x)$ Distribute the negative sign: $(f-g)(x) = 5x + 1 + 2 - 3x$ Combine like terms: $(f-g)(x) = 5x - 3x + 1 + 2 = 2x + 3$ So, $(f-g)(x) = 2x + 3$

Summary:

• $(f+g)(x) = 8x - 1$
• $(f-g)(x) = 2x + 3$

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 subtracting one function from another, carefully distribute the negative sign across all terms to avoid mistakes.