The function $f(x) = \frac{(-x+3)(x+4)}{x^2-9}$ can be analyzed as follows:

Step 1: Factorize the Denominator

The denominator $x^2 - 9$ is a difference of squares: $x^2 - 9 = (x - 3)(x + 3).$

Thus, the function becomes: $f(x) = \frac{(-x+3)(x+4)}{(x-3)(x+3)}.$

Step 2: Simplify the Function

The numerator has a factor of $(-x + 3)$, which simplifies to $-(x - 3)$. Substituting this: $f(x) = \frac{-(x-3)(x+4)}{(x-3)(x+3)}.$

The factor $(x-3)$ cancels in the numerator and denominator, except at $x = 3$, where the original function is undefined. This introduces a hole at $x = 3$.

The simplified function is: $f(x) = \frac{-(x+4)}{x+3}, \quad x \neq 3.$

Step 3: Identify Asymptotes and Holes

1. Vertical Asymptote: The denominator $x+3 = 0$ implies a vertical asymptote at: $x = -3.$

2. Hole: As noted earlier, there is a removable discontinuity (hole) at: $x = 3.$ To find the $y$-coordinate of the hole, substitute $x = 3$ into the simplified function: $f(x) = \frac{-(x+4)}{x+3}.$ At $x = 3$: $f(3) = \frac{-(3+4)}{3+3} = \frac{-7}{6}.$ So, the hole is at $(3, -\frac{7}{6})$.

3. Horizontal Asymptote: For large values of $x$, the highest-degree terms in the numerator and denominator dominate. The degree of the numerator and denominator are the same ($x^1$ in both), so the horizontal asymptote is the ratio of the leading coefficients: $y = \frac{-1}{1} = -1.$

Step 4: Graph the Function

• Plot the vertical asymptote at $x = -3$.
• Plot the horizontal asymptote at $y = -1$.
• Mark the hole at $(3, -\frac{7}{6})$.
• Sketch the behavior of the simplified function $f(x) = \frac{-(x+4)}{x+3}$, taking into account the asymptotes and hole.

Final Notes

Would you like me to generate a visual graph of this function? If yes, I can plot it for you. Let me know!

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Related Questions

1. How do you determine whether a function has a horizontal or oblique asymptote?
2. What are the steps to identify and classify discontinuities in a function?
3. Can a function have both removable discontinuities and vertical asymptotes? Provide an example.
4. How does factoring help simplify rational functions?
5. What is the significance of asymptotes in real-world applications of functions?

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Tip

Always check the original function for undefined points (such as $x$ values that make the denominator zero) before simplifying to avoid missing holes or discontinuities.