Let's break down the functions and operations step by step based on the provided image:

Given Functions:

• $f(x) = x^2 - 4$
• $g(x) = x^2 - 6x + 8$
• $h(x) = x - 4$

1. $(f + g)(x)$

This means we need to add $f(x)$ and $g(x)$: $(f + g)(x) = f(x) + g(x) = (x^2 - 4) + (x^2 - 6x + 8)$ $(f + g)(x) = x^2 + x^2 - 6x - 4 + 8 = 2x^2 - 6x + 4$

2. $(h - g)(x)$

This means we subtract $g(x)$ from $h(x)$: $(h - g)(x) = h(x) - g(x) = (x - 4) - (x^2 - 6x + 8)$ $(h - g)(x) = x - 4 - x^2 + 6x - 8 = -x^2 + 7x - 12$

3. $(f \times h)(x)$

This means we multiply $f(x)$ and $h(x)$: $(f \times h)(x) = f(x) \cdot h(x) = (x^2 - 4) \cdot (x - 4)$ Using the distributive property: $f(x) \cdot h(x) = x^2(x - 4) - 4(x - 4) = x^3 - 4x^2 - 4x + 16$ $(f \times h)(x) = x^3 - 4x^2 - 4x + 16$

4. $\left(\frac{g}{f}\right)(x)$

This means we divide $g(x)$ by $f(x)$: $\left(\frac{g}{f}\right)(x) = \frac{g(x)}{f(x)} = \frac{x^2 - 6x + 8}{x^2 - 4}$ Notice that $x^2 - 4 = (x - 2)(x + 2)$ and $x^2 - 6x + 8 = (x - 2)(x - 4)$. $\left(\frac{g}{f}\right)(x) = \frac{(x - 2)(x - 4)}{(x - 2)(x + 2)} = \frac{x - 4}{x + 2} \quad \text{(for } x \neq 2\text{)}$

5. $\left(\frac{h}{g}\right)(x)$

This means we divide $h(x)$ by $g(x)$: $\left(\frac{h}{g}\right)(x) = \frac{h(x)}{g(x)} = \frac{x - 4}{x^2 - 6x + 8}$ Since $x^2 - 6x + 8 = (x - 2)(x - 4)$, we have: $\left(\frac{h}{g}\right)(x) = \frac{x - 4}{(x - 2)(x - 4)} = \frac{1}{x - 2} \quad \text{(for } x \neq 2, 4\text{)}$

Do you want more details on any of these operations or have any questions?

Related Questions:

1. How do you simplify rational functions involving quadratic terms?
2. What is the process for multiplying two polynomials?
3. How do you find the common factors when simplifying rational expressions?
4. Can you explain how the distributive property works when multiplying polynomials?
5. How do you handle undefined points when simplifying rational expressions?

Tip: Always check for common factors before performing operations on rational expressions. This simplifies the process and helps avoid errors!