The problem given is as follows:

1. Given functions: $f(x) = 2x^2 - 5x + 3$ and $g(x) = x + 4$ Find:
   • $(f + g)(6)$
   • $(f \cdot g)(-2)$

Let's solve each part:

Part (a): $(f + g)(6)$

To find $(f + g)(6)$, we first calculate $f(6)$ and $g(6)$ and then add them:

1. Calculate $f(6)$: $f(6) = 2(6)^2 - 5(6) + 3 = 2 \cdot 36 - 30 + 3 = 72 - 30 + 3 = 45$

2. Calculate $g(6)$: $g(6) = 6 + 4 = 10$

3. Add $f(6)$ and $g(6)$: $(f + g)(6) = f(6) + g(6) = 45 + 10 = 55$

Thus, $(f + g)(6) = 55$.

Part (b): $(f \cdot g)(-2)$

To find $(f \cdot g)(-2)$, we first calculate $f(-2)$ and $g(-2)$, then multiply them:

1. Calculate $f(-2)$: $f(-2) = 2(-2)^2 - 5(-2) + 3 = 2 \cdot 4 + 10 + 3 = 8 + 10 + 3 = 21$

2. Calculate $g(-2)$: $g(-2) = -2 + 4 = 2$

3. Multiply $f(-2)$ and $g(-2)$: $(f \cdot g)(-2) = f(-2) \cdot g(-2) = 21 \cdot 2 = 42$

Thus, $(f \cdot g)(-2) = 42$.

Summary

• $(f + g)(6) = 55$
• $(f \cdot g)(-2) = 42$

Would you like further explanations on any of these steps?

Here are some related questions that might deepen your understanding:

1. How do you find the sum and product of two functions in general?
2. What are some properties of functions that help simplify calculations?
3. How would $(f - g)(x)$ and $(f / g)(x)$ be calculated for these functions?
4. What happens if you apply transformations to $f(x)$ and $g(x)$ before combining them?
5. Can you find the domain and range for the sum and product of $f(x)$ and $g(x)$?

Tip: When dealing with function operations, calculate each function individually at the given point before combining them for clarity.