To find the domain of each function $f(x)$ and $g(x)$, we need to identify the values of $x$ that make the denominator zero, as these values will be excluded from the domain.

1. Analyzing $f(x)$:

The function $f(x)$ is given by: $f(x) = \frac{x - 4}{x^2 - 16}$

1. Determine the denominator: The denominator is $x^2 - 16$.
2. Set the denominator equal to zero: $x^2 - 16 = 0$ This factors as: $(x - 4)(x + 4) = 0$ So, $x = 4$ and $x = -4$ are the values that make the denominator zero.
3. Domain of $f(x)$: Exclude $x = 4$ and $x = -4$ from the domain. Thus, the domain of $f(x)$ in interval notation is: $(-\infty, -4) \cup (-4, 4) \cup (4, \infty)$

2. Analyzing $g(x)$:

The function $g(x)$ is given by: $g(x) = \frac{x}{x^2 + 1}$

1. Determine the denominator: The denominator is $x^2 + 1$.
2. Set the denominator equal to zero: $x^2 + 1 = 0$ Solving for $x$, we get: $x^2 = -1$ Since $x^2 = -1$ has no real solutions, the denominator is never zero for real numbers.
3. Domain of $g(x)$: The function $g(x)$ is defined for all real values of $x$. Therefore, the domain of $g(x)$ is: $(-\infty, \infty)$

Summary:

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

Would you like a further breakdown of any steps?

Additional Questions

1. What would the range of each function be?
2. How would the domain change if we were dealing with complex numbers instead?
3. What is the behavior of $f(x)$ as $x$ approaches the excluded values?
4. Can $f(x)$ and $g(x)$ be simplified further?
5. How would the domain be affected if these functions were combined (e.g., $f(x) + g(x)$)?

Tip:

When determining the domain of a rational function, always start by identifying values that make the denominator zero, as these values are not included in the domain.