Welcome to Westonci.ca, the Q&A platform where your questions are met with detailed answers from experienced experts. Get detailed answers to your questions from a community of experts dedicated to providing accurate information. Join our platform to connect with experts ready to provide precise answers to your questions in different areas.
Sagot :
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]
Thank you for your visit. We are dedicated to helping you find the information you need, whenever you need it. Thank you for your visit. We're dedicated to helping you find the information you need, whenever you need it. Keep exploring Westonci.ca for more insightful answers to your questions. We're here to help.
