Thank you for pointing that out. Let's correct the calculations using the correct function \( f(x) = x^2 + 3 \). I'll go through each operation again.

### Given Functions:
- \( f(x) = x^2 + 3 \)
- \( g(x) = 10x - 9 \)

### 1. \( (f + g)(x) \)
\[ (f + g)(x) = f(x) + g(x) \]
Substitute the functions:
\[ (f + g)(x) = (x^2 + 3) + (10x - 9) \]
Simplify:
\[ (f + g)(x) = x^2 + 10x - 6 \]
- **Domain**: Both \( f(x) \) and \( g(x) \) are polynomials, which are defined for all real numbers.
- **Domain in interval notation**: \( (-\infty, \infty) \)

### 2. \( (f - g)(x) \)
\[ (f - g)(x) = f(x) - g(x) \]
Substitute the functions:
\[ (f - g)(x) = (x^2 + 3) - (10x - 9) \]
Simplify:
\[ (f - g)(x) = x^2 - 10x + 12 \]
- **Domain**: Since both \( f(x) \) and \( g(x) \) are defined for all real numbers, the domain is all real numbers.
- **Domain in interval notation**: \( (-\infty, \infty) \)

### 3. \( (f \cdot g)(x) \)
\[ (f \cdot g)(x) = f(x) \cdot g(x) \]
Substitute the functions:
\[ (f \cdot g)(x) = (x^2 + 3) \cdot (10x - 9) \]
Expand:
\[ (f \cdot g)(x) = 10x^3 - 9x^2 + 30x - 27 \]
- **Domain**: This is a polynomial, so it is defined for all real numbers.
- **Domain in interval notation**: \( (-\infty, \infty) \)

### 4. \( \left(\frac{f}{g}\right)(x) \)
\[ \left(\frac{f}{g}\right)(x) = \frac{f(x)}{g(x)} \]
Substitute the functions:
\[ \left(\frac{f}{g}\right)(x) = \frac{x^2 + 3}{10x - 9} \]
- **Domain**: The domain is all real numbers except where the denominator \( g(x) = 10x - 9 \) is zero.
- Set \( 10x - 9 = 0 \) and solve for \( x \):
\[ 10x = 9 \]
\[ x = \frac{9}{10} \]
- Therefore, the domain excludes \( x = \frac{9}{10} \).
- **Domain in interval notation**: \( (-\infty, \frac{9}{10}) \cup (\frac{9}{10}, \infty) \)

Please let me know if you have any questions or need further assistance!

---

Here are five related questions you might find helpful:
1. What is the difference in domain for a polynomial function compared to a rational function?
2. How would the operations differ if one of the functions involved a square root?
3. How would the domain change if \( f(x) = \frac{1}{x^2 + 3} \)?
4. Can you think of a scenario where the domain of \( (f \cdot g)(x) \) would not be all real numbers?
5. How do the operations on functions affect the range compared to the original functions?

**Tip**: Always double-check the expressions of the functions you're working with to avoid any errors in calculations, especially when simplifying expressions.

Explore how to add, subtract, multiply polynomials, and calculate domains for rational functions. Detailed solutions and domain considerations included.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation