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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.

Suppose that the functions g and h are defined for all real numbers x as follows: g(x) = 4x - 5, h(x) = x + 6. Write the expressions for (g + h)(x) and (g ⋅ h)(x) and evaluate (g − h)(3).

Learn how to perform function operations such as addition, multiplication, and evaluation with the functions g(x) = 4x - 5 and h(x) = x + 6. This problem involves key algebraic techniques and is suitable for students in grades 9-12.

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