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Let's solve the problem step by step.

Given:
- \( g(x) = 4x - 6 \)
- \( h(x) = x + 1 \)

We need to find the expressions for \( (h \cdot g)(x) \), \( (h - g)(x) \), and evaluate \( (h + g)(2) \).

### 1. Expression for \( (h \cdot g)(x) \)

\[
(h \cdot g)(x) = h(x) \cdot g(x)
\]

Substitute the given functions:

\[
(h \cdot g)(x) = (x + 1)(4x - 6)
\]

Now, expand this:

\[
(h \cdot g)(x) = 4x^2 - 6x + 4x - 6 = 4x^2 - 2x - 6
\]

### 2. Expression for \( (h - g)(x) \)

\[
(h - g)(x) = h(x) - g(x)
\]

Substitute the given functions:

\[
(h - g)(x) = (x + 1) - (4x - 6)
\]

Simplify:

\[
(h - g)(x) = x + 1 - 4x + 6 = -3x + 7
\]

### 3. Evaluate \( (h + g)(2) \)

\[
(h + g)(x) = h(x) + g(x)
\]

Substitute the given functions:

\[
(h + g)(x) = (x + 1) + (4x - 6)
\]

Simplify:

\[
(h + g)(x) = x + 1 + 4x - 6 = 5x - 5
\]

Now, substitute \( x = 2 \):

\[
(h + g)(2) = 5(2) - 5 = 10 - 5 = 5
\]

### Answers:

1. \( (h \cdot g)(x) = 4x^2 - 2x - 6 \)
2. \( (h - g)(x) = -3x + 7 \)
3. \( (h + g)(2) = 5 \)

Would you like further details or have any other questions?

### Related Questions:
1. What is the expression for \( (g \cdot h)(x) \)?
2. How would you find \( (g - h)(x) \)?
3. What is the value of \( (h - g)(0) \)?
4. How do you derive the expressions for composite functions like \( (g \circ h)(x) \)?
5. What is the derivative of \( (h \cdot g)(x) \)?

**Tip:** Always simplify expressions step-by-step to avoid errors and ensure clarity.

Write the expressions for (h · g)(x), (h - g)(x), and evaluate (h + g)(2). Given g(x) = 4x - 6 and h(x) = x + 1.

Learn how to perform operations on functions such as multiplying, subtracting, and adding functions, using the example with g(x) = 4x - 6 and h(x) = x + 1. This problem is suitable for students in Grades 9-12.

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