the functions f and g are defines as follows: f(x) = (x+3) / (x^2-6x+9) , g(x) = (x-7)(x^2-49), for each function, find the domain, write each answer as an interval or union of intervals

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.

Explore how to find the domain of rational functions and polynomial functions with detailed step-by-step solutions. This page covers the domain analysis of f(x) = (x+3)/(x^2 - 6x + 9) and g(x) = (x-7)(x^2 - 49), including identifying where the functions are undefined and using factorization methods. Suitable for advanced high school students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation