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.