The problem asks to find $g(x)$ given the following information:

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

Steps to solve for $g(x)$:

• By definition, $(f + g)(x) = f(x) + g(x)$.
• This means that: $f(x) + g(x) = 3x - 5$
• Substituting $f(x) = 2x + 3$ into the equation: $(2x + 3) + g(x) = 3x - 5$
• Solving for $g(x)$: $g(x) = (3x - 5) - (2x + 3)$ $g(x) = 3x - 5 - 2x - 3$ $g(x) = x - 8$

Thus, $g(x) = x - 8$.

Would you like a more detailed explanation, or do you have any questions?

Here are 5 related questions to deepen your understanding:

1. How would you solve for $g(x)$ if $f(x)$ had a quadratic form instead of linear?
2. What is the domain of $f(x)$ and $g(x)$ in this case?
3. Can you represent the sum of two functions graphically and how would this apply to $f(x) + g(x)$?
4. What happens to the combined function $f(x) + g(x)$ if you subtract a constant from both functions?
5. How would this change if you were given $(f \cdot g)(x)$ instead of $(f + g)(x)$?

Tip: When solving for a function in terms of others, carefully substitute given expressions and simplify step-by-step.