The problem in the image asks us to evaluate two expressions involving functions $f$ and $g$.

Given: $f(x) = 2x^2 - 5x + 3$ $g(x) = x + 4$

We are asked to find:

1. $(f + g)(6)$
2. $(f \cdot g)(-2)$

Solution

1. Finding $(f + g)(6)$:

   $(f + g)(x) = f(x) + g(x)$ Substitute $f(x) = 2x^2 - 5x + 3$ and $g(x) = x + 4$: $(f + g)(x) = (2x^2 - 5x + 3) + (x + 4)$ Simplify by combining like terms: $(f + g)(x) = 2x^2 - 4x + 7$ Now, substitute $x = 6$: $(f + g)(6) = 2(6)^2 - 4(6) + 7$ $= 2 \cdot 36 - 24 + 7$ $= 72 - 24 + 7$ $= 55$

2. Finding $(f \cdot g)(-2)$:

   $(f \cdot g)(x) = f(x) \cdot g(x)$ Substitute $f(x) = 2x^2 - 5x + 3$ and $g(x) = x + 4$: $(f \cdot g)(x) = (2x^2 - 5x + 3)(x + 4)$ We expand this expression: $(f \cdot g)(x) = 2x^3 + 8x^2 - 5x^2 - 20x + 3x + 12$ Combine like terms: $(f \cdot g)(x) = 2x^3 + 3x^2 - 17x + 12$ Now, substitute $x = -2$: $(f \cdot g)(-2) = 2(-2)^3 + 3(-2)^2 - 17(-2) + 12$ $= 2(-8) + 3(4) + 34 + 12$ $= -16 + 12 + 34 + 12$ $= 42$

Final Answers:

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

Would you like further details on these calculations or explanations?

Here are some related questions to deepen understanding:

1. How do you find the sum of two functions in general?
2. How can you simplify the product of two functions?
3. What is the significance of evaluating a function at a specific point?
4. How would the process change if $g(x)$ were a quadratic function?
5. How do you apply polynomial multiplication in function composition?

Tip: Always combine like terms after expanding to simplify expressions efficiently.