Question: User: f(x)= 3x 2 +8x x 2 −9
1. **State the problem:** We are given the function $$f(x) = \frac{3x^2 + 8x}{x^2 - 9}$$ and we want to understand or simplify it.
2. **Recall the formula and rules:** The function is a rational function, which is a fraction where both numerator and denominator are polynomials.
3. **Factor the denominator:** Notice that $$x^2 - 9$$ is a difference of squares, so it factors as:
$$x^2 - 9 = (x - 3)(x + 3)$$
4. **Factor the numerator if possible:** The numerator is $$3x^2 + 8x$$. We can factor out an $$x$$:
$$3x^2 + 8x = x(3x + 8)$$
5. **Rewrite the function with factored forms:**
$$f(x) = \frac{x(3x + 8)}{(x - 3)(x + 3)}$$
6. **Check for common factors:** There are no common factors between numerator and denominator, so the function is simplified as is.
7. **Domain considerations:** The denominator cannot be zero, so $$x \neq 3$$ and $$x \neq -3$$.
**Final simplified form:**
$$f(x) = \frac{x(3x + 8)}{(x - 3)(x + 3)}$$