Let's find the domain of each function based on the given definitions.

1. Domain of $f(x) = \frac{x^2}{x + 4}$

The domain of a rational function is all real numbers except where the denominator equals zero, as division by zero is undefined. Here, the denominator is $x + 4$.

Set the denominator equal to zero: $x + 4 = 0$ $x = -4$ So, the function $f(x)$ is undefined at $x = -4$. Therefore, the domain of $f(x)$ is all real numbers except $-4$.

Domain of $f(x)$: $(-\infty, -4) \cup (-4, \infty)$

2. Domain of $g(x) = \frac{x + 7}{x^2 - 49}$

Again, the domain of a rational function is all real numbers except where the denominator is zero. Here, the denominator is $x^2 - 49$, which can be factored as: $x^2 - 49 = (x - 7)(x + 7)$ Set the denominator equal to zero: $(x - 7)(x + 7) = 0$ This gives $x = 7$ and $x = -7$, meaning the function $g(x)$ is undefined at these points.

Domain of $g(x)$: $(-\infty, -7) \cup (-7, 7) \cup (7, \infty)$

Would you like further details or have any questions on these domain calculations?

Related questions:

1. How do you handle functions with more complex denominators?
2. What are the general steps to finding the domain of a composite function?
3. How would you find the range of the functions $f(x)$ and $g(x)$?
4. What happens to the domain when there are square roots involved?
5. Can vertical asymptotes be inferred from the domain restrictions?

Tip:

When finding the domain of a rational function, always look for values that make the denominator zero, and exclude those from the domain.