The image shows a problem that involves finding the domain of two rational functions. Let’s analyze and solve the problem step-by-step:

Given Functions

1. $f(x) = \frac{x^2}{x^2 - 9}$
2. $g(x) = \frac{x}{x - 3}$

Domain of a Function

The domain of a function includes all real numbers except where the function is undefined. For rational functions, these points occur where the denominator is zero.

Step-by-step Solutions

1. Domain of $f(x) = \frac{x^2}{x^2 - 9}$

• The denominator is $x^2 - 9$.
• Set the denominator equal to zero:
  $x^2 - 9 = 0$ $x^2 = 9$ $x = \pm 3$
• Therefore, the function $f(x)$ is undefined at $x = 3$ and $x = -3$.

Domain of $f(x)$: All real numbers except $x = 3$ and $x = -3$.
In interval notation: $(-\infty, -3) \cup (-3, 3) \cup (3, \infty)$.

2. Domain of $g(x) = \frac{x}{x - 3}$

• The denominator is $x - 3$.
• Set the denominator equal to zero:
  $x - 3 = 0$ $x = 3$
• Therefore, the function $g(x)$ is undefined at $x = 3$.

Domain of $g(x)$: All real numbers except $x = 3$.
In interval notation: $(-\infty, 3) \cup (3, \infty)$.

Conclusion

• Domain of $f(x)$: $(-\infty, -3) \cup (-3, 3) \cup (3, \infty)$
• Domain of $g(x)$: $(-\infty, 3) \cup (3, \infty)$

Would you like more details or have any questions?

Related Questions to Explore:

1. How do you find the domain of a function with both a numerator and denominator that are polynomials?
2. What happens to the graph of a function at points where it is undefined?
3. How do you find the range of a rational function?
4. What is the difference between a vertical asymptote and a hole in the graph of a function?
5. How does the concept of limits help in understanding points where a function is undefined?

Tip

When determining the domain of a rational function, always identify the values that make the denominator zero and exclude them from the domain.