To find the domain of the functions $f(x)$ and $g(x)$, we need to identify the values of $x$ for which each function is defined. This typically involves finding where the denominator is zero (for rational functions) or where the expression is undefined (for polynomials).

Function $f(x) = \frac{x+3}{x^2 - 6x + 9}$

1. Identify the denominator: $x^2 - 6x + 9$.

2. Set the denominator equal to zero and solve for $x$: $x^2 - 6x + 9 = 0$ Factor the quadratic expression: $(x - 3)(x - 3) = 0$ So, $x = 3$.

   This means $f(x)$ is undefined when $x = 3$.

3. Domain of $f(x)$: The function is defined for all real numbers except $x = 3$.

   • In interval notation, the domain of $f(x)$ is $\boxed{(-\infty, 3) \cup (3, \infty)}$.

Function $g(x) = (x-7)(x^2 - 49)$

1. Simplify the expression: Note that $x^2 - 49$ can be factored as $(x - 7)(x + 7)$. So, $g(x) = (x-7)(x-7)(x+7) = (x-7)^2(x+7)$

2. Identify any restrictions: Since $g(x)$ is a polynomial function, it is defined for all real values of $x$. There are no restrictions.

3. Domain of $g(x)$: The domain of $g(x)$ is all real numbers.

   • In interval notation, the domain of $g(x)$ is $\boxed{(-\infty, \infty)}$.

Summary

• The domain of $f(x) = \frac{x+3}{x^2 - 6x + 9}$ is $(-\infty, 3) \cup (3, \infty)$.
• The domain of $g(x) = (x-7)(x^2 - 49)$ is $(-\infty, \infty)$.

Would you like further details or have any questions?

Relative Questions:

1. How do you determine the domain of a rational function?
2. What is the process for factoring quadratic expressions?
3. How do you identify when a polynomial function is defined for all real numbers?
4. What are the implications of having a function with a domain of all real numbers?
5. How does the presence of a square in the denominator affect the domain of a function?

Tip:

When factoring a quadratic expression, always check for perfect square trinomials, as they simplify the factorization process significantly.