Answer :
To find the functions [tex]\(f(x) + g(x)\)[/tex], [tex]\(f(x) - g(x)\)[/tex], [tex]\(f(x) \cdot g(x)\)[/tex], and [tex]\(\frac{f(x)}{g(x)}\)[/tex], and to determine their domains, follow these steps:
### Given Functions:
[tex]\[ f(x) = 4x - 5 \][/tex]
[tex]\[ g(x) = x + 6 \][/tex]
### 1. Finding [tex]\( (f+g)(x) \)[/tex]:
Combine [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex]:
[tex]\[ (f + g)(x) = f(x) + g(x) \][/tex]
[tex]\[ (f + g)(x) = (4x - 5) + (x + 6) \][/tex]
Simplify:
[tex]\[ (f + g)(x) = 4x - 5 + x + 6 \][/tex]
[tex]\[ (f + g)(x) = 5x + 1 \][/tex]
### 2. Finding [tex]\( (f-g)(x) \)[/tex]:
Subtract [tex]\( g(x) \)[/tex] from [tex]\( f(x) \)[/tex]:
[tex]\[ (f - g)(x) = f(x) - g(x) \][/tex]
[tex]\[ (f - g)(x) = (4x - 5) - (x + 6) \][/tex]
Simplify:
[tex]\[ (f - g)(x) = 4x - 5 - x - 6 \][/tex]
[tex]\[ (f - g)(x) = 3x - 11 \][/tex]
### 3. Finding [tex]\( (f \cdot g)(x) \)[/tex]:
Multiply [tex]\( f(x) \)[/tex] by [tex]\( g(x) \)[/tex]:
[tex]\[ (f \cdot g)(x) = f(x) \cdot g(x) \][/tex]
[tex]\[ (f \cdot g)(x) = (4x - 5) \cdot (x + 6) \][/tex]
Apply the distributive property:
[tex]\[ (f \cdot g)(x) = 4x(x + 6) - 5(x + 6) \][/tex]
[tex]\[ (f \cdot g)(x) = 4x^2 + 24x - 5x - 30 \][/tex]
Combine like terms:
[tex]\[ (f \cdot g)(x) = 4x^2 + 19x - 30 \][/tex]
### 4. Finding [tex]\( \left(\frac{f}{g}\right)(x) \)[/tex]:
Divide [tex]\( f(x) \)[/tex] by [tex]\( g(x) \)[/tex]:
[tex]\[ \left(\frac{f}{g}\right)(x) = \frac{f(x)}{g(x)} \][/tex]
[tex]\[ \left(\frac{f}{g}\right)(x) = \frac{4x - 5}{x + 6} \][/tex]
### Domains:
- The domain of [tex]\(f(x) = 4x - 5\)[/tex] is all real numbers.
- The domain of [tex]\(g(x) = x + 6\)[/tex] is all real numbers.
Thus:
- The domain of [tex]\( (f + g)(x) = 5x + 1\)[/tex] is all real numbers.
- The domain of [tex]\( (f - g)(x) = 3x - 11\)[/tex] is all real numbers.
- The domain of [tex]\( (f \cdot g)(x) = 4x^2 + 19x - 30\)[/tex] is all real numbers.
- The domain of [tex]\( \left(\frac{f}{g}\right)(x) = \frac{4x - 5}{x + 6} \)[/tex] is all real numbers except [tex]\( x = -6 \)[/tex], because at [tex]\( x = -6 \)[/tex], the denominator [tex]\( g(x) \)[/tex] becomes zero, which makes the function undefined.
### Summary:
[tex]\[ (f + g)(x) = 5x + 1 \][/tex] \\
[tex]\[ (f - g)(x) = 3x - 11 \][/tex] \\
[tex]\[ (f \cdot g)(x) = 4x^2 + 19x - 30 \][/tex] \\
[tex]\[ \left(\frac{f}{g}\right)(x) = \frac{4x - 5}{x + 6} \][/tex] \\
Domains:
- [tex]\( (f + g)(x) \)[/tex]: All real numbers
- [tex]\( (f - g)(x) \)[/tex]: All real numbers
- [tex]\( (f \cdot g)(x) \)[/tex]: All real numbers
- [tex]\( \left(\frac{f}{g}\right)(x) \)[/tex]: All real numbers except [tex]\( x = -6 \)[/tex]
### Given Functions:
[tex]\[ f(x) = 4x - 5 \][/tex]
[tex]\[ g(x) = x + 6 \][/tex]
### 1. Finding [tex]\( (f+g)(x) \)[/tex]:
Combine [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex]:
[tex]\[ (f + g)(x) = f(x) + g(x) \][/tex]
[tex]\[ (f + g)(x) = (4x - 5) + (x + 6) \][/tex]
Simplify:
[tex]\[ (f + g)(x) = 4x - 5 + x + 6 \][/tex]
[tex]\[ (f + g)(x) = 5x + 1 \][/tex]
### 2. Finding [tex]\( (f-g)(x) \)[/tex]:
Subtract [tex]\( g(x) \)[/tex] from [tex]\( f(x) \)[/tex]:
[tex]\[ (f - g)(x) = f(x) - g(x) \][/tex]
[tex]\[ (f - g)(x) = (4x - 5) - (x + 6) \][/tex]
Simplify:
[tex]\[ (f - g)(x) = 4x - 5 - x - 6 \][/tex]
[tex]\[ (f - g)(x) = 3x - 11 \][/tex]
### 3. Finding [tex]\( (f \cdot g)(x) \)[/tex]:
Multiply [tex]\( f(x) \)[/tex] by [tex]\( g(x) \)[/tex]:
[tex]\[ (f \cdot g)(x) = f(x) \cdot g(x) \][/tex]
[tex]\[ (f \cdot g)(x) = (4x - 5) \cdot (x + 6) \][/tex]
Apply the distributive property:
[tex]\[ (f \cdot g)(x) = 4x(x + 6) - 5(x + 6) \][/tex]
[tex]\[ (f \cdot g)(x) = 4x^2 + 24x - 5x - 30 \][/tex]
Combine like terms:
[tex]\[ (f \cdot g)(x) = 4x^2 + 19x - 30 \][/tex]
### 4. Finding [tex]\( \left(\frac{f}{g}\right)(x) \)[/tex]:
Divide [tex]\( f(x) \)[/tex] by [tex]\( g(x) \)[/tex]:
[tex]\[ \left(\frac{f}{g}\right)(x) = \frac{f(x)}{g(x)} \][/tex]
[tex]\[ \left(\frac{f}{g}\right)(x) = \frac{4x - 5}{x + 6} \][/tex]
### Domains:
- The domain of [tex]\(f(x) = 4x - 5\)[/tex] is all real numbers.
- The domain of [tex]\(g(x) = x + 6\)[/tex] is all real numbers.
Thus:
- The domain of [tex]\( (f + g)(x) = 5x + 1\)[/tex] is all real numbers.
- The domain of [tex]\( (f - g)(x) = 3x - 11\)[/tex] is all real numbers.
- The domain of [tex]\( (f \cdot g)(x) = 4x^2 + 19x - 30\)[/tex] is all real numbers.
- The domain of [tex]\( \left(\frac{f}{g}\right)(x) = \frac{4x - 5}{x + 6} \)[/tex] is all real numbers except [tex]\( x = -6 \)[/tex], because at [tex]\( x = -6 \)[/tex], the denominator [tex]\( g(x) \)[/tex] becomes zero, which makes the function undefined.
### Summary:
[tex]\[ (f + g)(x) = 5x + 1 \][/tex] \\
[tex]\[ (f - g)(x) = 3x - 11 \][/tex] \\
[tex]\[ (f \cdot g)(x) = 4x^2 + 19x - 30 \][/tex] \\
[tex]\[ \left(\frac{f}{g}\right)(x) = \frac{4x - 5}{x + 6} \][/tex] \\
Domains:
- [tex]\( (f + g)(x) \)[/tex]: All real numbers
- [tex]\( (f - g)(x) \)[/tex]: All real numbers
- [tex]\( (f \cdot g)(x) \)[/tex]: All real numbers
- [tex]\( \left(\frac{f}{g}\right)(x) \)[/tex]: All real numbers except [tex]\( x = -6 \)[/tex]
