To determine which function $g(x)$ could represent the rational function with a vertical asymptote at $x = -3$, we need to examine each option and check if the denominator has a factor of $(x + 3)$, which would cause a vertical asymptote at $x = -3$.

Let’s go through each option to analyze the denominator:

1. Option 1: $g(x) = \frac{x^2 + 9x + 20}{x^2 - x - 12}$

   • Factor the denominator: $x^2 - x - 12 = (x - 4)(x + 3)$.
   • This has a factor of $(x + 3)$, so it will have a vertical asymptote at $x = -3$.

2. Option 2: $g(x) = \frac{x^2 + 7x + 12}{x^2 - x - 20}$

   • Factor the denominator: $x^2 - x - 20 = (x - 5)(x + 4)$.
   • This does not have a factor of $(x + 3)$, so it does not produce a vertical asymptote at $x = -3$.

3. Option 3: $g(x) = \frac{x^2 - 2x - 15}{x^2 + x - 20}$

   • Factor the denominator: $x^2 + x - 20 = (x - 4)(x + 5)$.
   • This does not have a factor of $(x + 3)$, so it does not produce a vertical asymptote at $x = -3$.

4. Option 4: $g(x) = \frac{x^2 + 8x + 15}{x^2 - 7x + 12}$

   • Factor the denominator: $x^2 - 7x + 12 = (x - 3)(x - 4)$.
   • This does not have a factor of $(x + 3)$, so it does not produce a vertical asymptote at $x = -3$.

Conclusion

The only option that has a vertical asymptote at $x = -3$ is Option 1: $g(x) = \frac{x^2 + 9x + 20}{x^2 - x - 12}$.

Would you like further explanation on vertical asymptotes or factoring?

Related Questions

1. How do vertical asymptotes affect the graph of a rational function?
2. Why does a factor in the denominator cause a vertical asymptote?
3. What other types of asymptotes can a rational function have?
4. How would you find horizontal asymptotes for these functions?
5. Can a rational function have more than one vertical asymptote?

Tip

When looking for vertical asymptotes, focus on the roots of the denominator, as these values make the function undefined and create potential asymptotes.