To identify the rule for $f + g$, we add the given functions $f(x)$ and $g(x)$.

Given:

• $f(x) = 3x^2 + 5x + 1$
• $g(x) = 2x - 1$

Add $f(x)$ and $g(x)$:

$(f + g)(x) = f(x) + g(x) = (3x^2 + 5x + 1) + (2x - 1)$

Combine like terms: $(f + g)(x) = 3x^2 + (5x + 2x) + (1 - 1)$

Simplify: $(f + g)(x) = 3x^2 + 7x$

Final Answer:

The rule for $f + g$ is: $(f + g)(x) = 3x^2 + 7x$

Would you like further details or have any questions?

Additional Questions:

1. What is the rule for $f - g$?
2. How do you evaluate $(f + g)(2)$?
3. What is the degree of the resulting polynomial $f + g$?
4. How do the leading coefficients of $f(x)$ and $g(x)$ affect $f + g$?
5. What happens to $f + g$ if $g(x) = -f(x)$?

Tip:

When combining functions, always align like terms (same powers of $x$) to simplify correctly.