Let's solve the given problem step by step.

We are given the following functions:

$g(x) = 4x - 5$ $h(x) = x + 6$

We need to:

1. Find $(g + h)(x)$
2. Find $(g \cdot h)(x)$
3. Evaluate $(g - h)(3)$

1. $(g + h)(x)$:

This is the sum of $g(x)$ and $h(x)$:

$(g + h)(x) = g(x) + h(x) = (4x - 5) + (x + 6)$ Simplifying: $(g + h)(x) = 4x - 5 + x + 6 = 5x + 1$

2. $(g \cdot h)(x)$:

This is the product of $g(x)$ and $h(x)$:

$(g \cdot h)(x) = g(x) \cdot h(x) = (4x - 5)(x + 6)$ Using the distributive property (FOIL): $(g \cdot h)(x) = 4x(x + 6) - 5(x + 6) = 4x^2 + 24x - 5x - 30$ Simplifying: $(g \cdot h)(x) = 4x^2 + 19x - 30$

3. $(g - h)(3)$:

This is the difference between $g(3)$ and $h(3)$. Let's first find $g(3)$ and $h(3)$:

$g(3) = 4(3) - 5 = 12 - 5 = 7$ $h(3) = 3 + 6 = 9$

Now, subtract: $(g - h)(3) = g(3) - h(3) = 7 - 9 = -2$

Final Answers:

1. $(g + h)(x) = 5x + 1$
2. $(g \cdot h)(x) = 4x^2 + 19x - 30$
3. $(g - h)(3) = -2$

Would you like further details or have any other questions?

Here are 5 related questions to explore further:

1. What is $(g/h)(x)$ for these functions?
2. How would you find $(g^2 + h^2)(x)$?
3. Can you find the domain of the function $g/h(x)$?
4. What is the derivative of $(g \cdot h)(x)$?
5. How would you graph the functions $g(x)$ and $h(x)$ together?

Tip: Always simplify expressions before evaluating specific values to avoid errors during calculations.