Answer :
To find the required functions and their domains, we need to perform basic operations on the given functions [tex]\( f(x) = 3x - 8 \)[/tex] and [tex]\( g(x) = x - 3 \)[/tex].
### 1. Addition: [tex]\( (f + g)(x) \)[/tex]
First, add the functions:
[tex]\[ f(x) + g(x) = (3x - 8) + (x - 3) \][/tex]
Combine like terms:
[tex]\[ (3x + x) - (8 + 3) = 4x - 11 \][/tex]
Thus,
[tex]\[ (f + g)(x) = 4x - 11 \][/tex]
The domain of the sum [tex]\( (f + g)(x) \)[/tex] is all real numbers, denoted as [tex]\( \mathbb{R} \)[/tex], since the sum of two polynomials is a polynomial, and polynomials are defined for all real numbers.
### 2. Subtraction: [tex]\( (f - g)(x) \)[/tex]
Next, subtract the functions:
[tex]\[ f(x) - g(x) = (3x - 8) - (x - 3) \][/tex]
Distribute the minus sign and combine like terms:
[tex]\[ (3x - x) - (8 - 3) = 2x - 5 \][/tex]
Thus,
[tex]\[ (f - g)(x) = 2x - 5 \][/tex]
The domain of the difference [tex]\( (f - g)(x) \)[/tex] is also all real numbers, [tex]\( \mathbb{R} \)[/tex], since it is also a polynomial.
### 3. Multiplication: [tex]\( (fg)(x) \)[/tex]
Multiply the functions:
[tex]\[ f(x) \cdot g(x) = (3x - 8)(x - 3) \][/tex]
Distribute the terms:
[tex]\[ = 3x \cdot x - 3x \cdot 3 - 8 \cdot x + 8 \cdot 3 \][/tex]
[tex]\[ = 3x^2 - 9x - 8x + 24 \][/tex]
Combine like terms:
[tex]\[ = 3x^2 - 17x + 24 \][/tex]
Thus,
[tex]\[ (fg)(x) = (3x - 8)(x - 3) \][/tex]
The domain of the product [tex]\( (fg)(x) \)[/tex] is all real numbers, [tex]\( \mathbb{R} \)[/tex], since it is a polynomial.
### 4. Division: [tex]\( \left(\frac{f}{g}\right)(x) \)[/tex]
Finally, divide the functions:
[tex]\[ \left(\frac{f}{g}\right)(x) = \frac{3x - 8}{x - 3} \][/tex]
The expression [tex]\( \frac{f}{g} \)[/tex] takes all real numbers except where the denominator is zero. Set the denominator equal to zero and solve for [tex]\( x \)[/tex]:
[tex]\[ x - 3 = 0 \][/tex]
[tex]\[ x = 3 \][/tex]
So, [tex]\( x = 3 \)[/tex] is not in the domain of [tex]\( \left(\frac{f}{g}\right)(x) \)[/tex]. Thus, the domain is all real numbers except [tex]\( x = 3 \)[/tex], denoted as:
[tex]\[ \mathbb{R} \setminus \{3\} \quad \text{or} \quad (-\infty, 3) \cup (3, \infty) \][/tex]
In summary:
[tex]\[ (f + g)(x) = 4x - 11 \quad \text{with domain} \quad \mathbb{R} \][/tex]
[tex]\[ (f - g)(x) = 2x - 5 \quad \text{with domain} \quad \mathbb{R} \][/tex]
[tex]\[ (fg)(x) = (3x - 8)(x - 3) \quad \text{with domain} \quad \mathbb{R} \][/tex]
[tex]\[ \left(\frac{f}{g}\right)(x) = \frac{3x - 8}{x - 3} \quad \text{with domain} \quad \mathbb{R} \setminus \{3\} \][/tex]
### 1. Addition: [tex]\( (f + g)(x) \)[/tex]
First, add the functions:
[tex]\[ f(x) + g(x) = (3x - 8) + (x - 3) \][/tex]
Combine like terms:
[tex]\[ (3x + x) - (8 + 3) = 4x - 11 \][/tex]
Thus,
[tex]\[ (f + g)(x) = 4x - 11 \][/tex]
The domain of the sum [tex]\( (f + g)(x) \)[/tex] is all real numbers, denoted as [tex]\( \mathbb{R} \)[/tex], since the sum of two polynomials is a polynomial, and polynomials are defined for all real numbers.
### 2. Subtraction: [tex]\( (f - g)(x) \)[/tex]
Next, subtract the functions:
[tex]\[ f(x) - g(x) = (3x - 8) - (x - 3) \][/tex]
Distribute the minus sign and combine like terms:
[tex]\[ (3x - x) - (8 - 3) = 2x - 5 \][/tex]
Thus,
[tex]\[ (f - g)(x) = 2x - 5 \][/tex]
The domain of the difference [tex]\( (f - g)(x) \)[/tex] is also all real numbers, [tex]\( \mathbb{R} \)[/tex], since it is also a polynomial.
### 3. Multiplication: [tex]\( (fg)(x) \)[/tex]
Multiply the functions:
[tex]\[ f(x) \cdot g(x) = (3x - 8)(x - 3) \][/tex]
Distribute the terms:
[tex]\[ = 3x \cdot x - 3x \cdot 3 - 8 \cdot x + 8 \cdot 3 \][/tex]
[tex]\[ = 3x^2 - 9x - 8x + 24 \][/tex]
Combine like terms:
[tex]\[ = 3x^2 - 17x + 24 \][/tex]
Thus,
[tex]\[ (fg)(x) = (3x - 8)(x - 3) \][/tex]
The domain of the product [tex]\( (fg)(x) \)[/tex] is all real numbers, [tex]\( \mathbb{R} \)[/tex], since it is a polynomial.
### 4. Division: [tex]\( \left(\frac{f}{g}\right)(x) \)[/tex]
Finally, divide the functions:
[tex]\[ \left(\frac{f}{g}\right)(x) = \frac{3x - 8}{x - 3} \][/tex]
The expression [tex]\( \frac{f}{g} \)[/tex] takes all real numbers except where the denominator is zero. Set the denominator equal to zero and solve for [tex]\( x \)[/tex]:
[tex]\[ x - 3 = 0 \][/tex]
[tex]\[ x = 3 \][/tex]
So, [tex]\( x = 3 \)[/tex] is not in the domain of [tex]\( \left(\frac{f}{g}\right)(x) \)[/tex]. Thus, the domain is all real numbers except [tex]\( x = 3 \)[/tex], denoted as:
[tex]\[ \mathbb{R} \setminus \{3\} \quad \text{or} \quad (-\infty, 3) \cup (3, \infty) \][/tex]
In summary:
[tex]\[ (f + g)(x) = 4x - 11 \quad \text{with domain} \quad \mathbb{R} \][/tex]
[tex]\[ (f - g)(x) = 2x - 5 \quad \text{with domain} \quad \mathbb{R} \][/tex]
[tex]\[ (fg)(x) = (3x - 8)(x - 3) \quad \text{with domain} \quad \mathbb{R} \][/tex]
[tex]\[ \left(\frac{f}{g}\right)(x) = \frac{3x - 8}{x - 3} \quad \text{with domain} \quad \mathbb{R} \setminus \{3\} \][/tex]
