IDNLearn.com: Your trusted source for accurate and reliable answers. Get prompt and accurate answers to your questions from our experts who are always ready to help.
Sagot :
To solve for the expressions [tex]\( f+g \)[/tex], [tex]\( f-g \)[/tex], [tex]\( f \cdot g \)[/tex], and [tex]\(\frac{f}{g}\)[/tex] given the functions [tex]\( f(x)=3x^2 + 25x + 42 \)[/tex] and [tex]\( g(x)=x + 6 \)[/tex], we need to perform algebraic operations on these functions.
### 1. Finding [tex]\((f+g)(x)\)[/tex]:
[tex]\[ (f + g)(x) = f(x) + g(x) \][/tex]
[tex]\[ f(x) = 3x^2 + 25x + 42 \][/tex]
[tex]\[ g(x) = x + 6 \][/tex]
Adding the two functions together:
[tex]\[ (f + g)(x) = (3x^2 + 25x + 42) + (x + 6) \][/tex]
Combine like terms:
[tex]\[ (f + g)(x) = 3x^2 + 25x + x + 42 + 6 \][/tex]
[tex]\[ (f + g)(x) = 3x^2 + 26x + 48 \][/tex]
So, [tex]\((f+g)(x) = 3x^2 + 26x + 48\)[/tex].
### 2. Finding [tex]\((f-g)(x)\)[/tex]:
[tex]\[ (f - g)(x) = f(x) - g(x) \][/tex]
[tex]\[ f(x) = 3x^2 + 25x + 42 \][/tex]
[tex]\[ g(x) = x + 6 \][/tex]
Subtracting the second function from the first:
[tex]\[ (f - g)(x) = (3x^2 + 25x + 42) - (x + 6) \][/tex]
Combine like terms:
[tex]\[ (f - g)(x) = 3x^2 + 25x - x + 42 - 6 \][/tex]
[tex]\[ (f - g)(x) = 3x^2 + 24x + 36 \][/tex]
So, [tex]\((f-g)(x) = 3x^2 + 24x + 36\)[/tex].
### 3. Finding [tex]\(f \cdot g\)[/tex] (product of [tex]\( f \)[/tex] and [tex]\( g \)[/tex]):
[tex]\[ (f \cdot g)(x) = f(x) \cdot g(x) \][/tex]
[tex]\[ f(x) = 3x^2 + 25x + 42 \][/tex]
[tex]\[ g(x) = x + 6 \][/tex]
Multiplying the two functions:
[tex]\[ (f \cdot g)(x) = (3x^2 + 25x + 42) \cdot (x + 6) \][/tex]
Apply the distributive property:
[tex]\[ (f \cdot g)(x) = 3x^2(x + 6) + 25x(x + 6) + 42(x + 6) \][/tex]
[tex]\[ (f \cdot g)(x) = 3x^3 + 18x^2 + 25x^2 + 150x + 42x + 252 \][/tex]
Combine like terms:
[tex]\[ (f \cdot g)(x) = 3x^3 + (18x^2 + 25x^2) + (150x + 42x) + 252 \][/tex]
[tex]\[ (f \cdot g)(x) = 3x^3 + 43x^2 + 192x + 252 \][/tex]
So, [tex]\( (f \cdot g)(x) = 3x^3 + 43x^2 + 192x + 252 \)[/tex].
### 4. Finding [tex]\(\frac{f}{g}\)[/tex] (quotient of [tex]\( f \)[/tex] and [tex]\( g \)[/tex]):
[tex]\[ \left( \frac{f}{g} \right)(x) = \frac{f(x)}{g(x)} \][/tex]
[tex]\[ f(x) = 3x^2 + 25x + 42 \][/tex]
[tex]\[ g(x) = x + 6 \][/tex]
So,
[tex]\[ \left( \frac{f}{g} \right)(x) = \frac{3x^2 + 25x + 42}{x + 6} \][/tex]
### Determining the Domains:
1. Domain of [tex]\((f+g)(x) = 3x^2 + 26x + 48\)[/tex]:
- This is a polynomial function, and polynomial functions are defined for all real numbers.
- Domain: [tex]\(\mathbb{R}\)[/tex] (all real numbers).
2. Domain of [tex]\((f-g)(x) = 3x^2 + 24x + 36\)[/tex]:
- This is also a polynomial function, defined for all real numbers.
- Domain: [tex]\(\mathbb{R}\)[/tex] (all real numbers).
3. Domain of [tex]\( (f \cdot g)(x) = 3x^3 + 43x^2 + 192x + 252 \)[/tex]:
- This is a polynomial function, defined for all real numbers.
- Domain: [tex]\(\mathbb{R}\)[/tex] (all real numbers).
4. Domain of [tex]\(\left( \frac{f}{g} \right)(x) = \frac{3x^2 + 25x + 42}{x + 6} \)[/tex]:
- The rational function is undefined when the denominator is zero.
- Set the denominator to zero to find the points to exclude:
[tex]\[ x + 6 = 0 \][/tex]
[tex]\[ x = -6 \][/tex]
Hence, the function is undefined at [tex]\( x = -6 \)[/tex].
- Domain: [tex]\(\mathbb{R} \setminus \{-6\}\)[/tex] (all real numbers except [tex]\( -6 \)[/tex]).
### Summary:
- [tex]\( (f+g)(x) = 3x^2 + 26x + 48 \)[/tex]
- Domain: [tex]\(\mathbb{R}\)[/tex]
- [tex]\( (f-g)(x) = 3x^2 + 24x + 36 \)[/tex]
- Domain: [tex]\(\mathbb{R}\)[/tex]
- [tex]\( (f \cdot g)(x) = 3x^3 + 43x^2 + 192x + 252 \)[/tex]
- Domain: [tex]\(\mathbb{R}\)[/tex]
- [tex]\( \left( \frac{f}{g} \right)(x) = \frac{3x^2 + 25x + 42}{x + 6} \)[/tex]
- Domain: [tex]\(\mathbb{R} \setminus \{-6\}\)[/tex]
### 1. Finding [tex]\((f+g)(x)\)[/tex]:
[tex]\[ (f + g)(x) = f(x) + g(x) \][/tex]
[tex]\[ f(x) = 3x^2 + 25x + 42 \][/tex]
[tex]\[ g(x) = x + 6 \][/tex]
Adding the two functions together:
[tex]\[ (f + g)(x) = (3x^2 + 25x + 42) + (x + 6) \][/tex]
Combine like terms:
[tex]\[ (f + g)(x) = 3x^2 + 25x + x + 42 + 6 \][/tex]
[tex]\[ (f + g)(x) = 3x^2 + 26x + 48 \][/tex]
So, [tex]\((f+g)(x) = 3x^2 + 26x + 48\)[/tex].
### 2. Finding [tex]\((f-g)(x)\)[/tex]:
[tex]\[ (f - g)(x) = f(x) - g(x) \][/tex]
[tex]\[ f(x) = 3x^2 + 25x + 42 \][/tex]
[tex]\[ g(x) = x + 6 \][/tex]
Subtracting the second function from the first:
[tex]\[ (f - g)(x) = (3x^2 + 25x + 42) - (x + 6) \][/tex]
Combine like terms:
[tex]\[ (f - g)(x) = 3x^2 + 25x - x + 42 - 6 \][/tex]
[tex]\[ (f - g)(x) = 3x^2 + 24x + 36 \][/tex]
So, [tex]\((f-g)(x) = 3x^2 + 24x + 36\)[/tex].
### 3. Finding [tex]\(f \cdot g\)[/tex] (product of [tex]\( f \)[/tex] and [tex]\( g \)[/tex]):
[tex]\[ (f \cdot g)(x) = f(x) \cdot g(x) \][/tex]
[tex]\[ f(x) = 3x^2 + 25x + 42 \][/tex]
[tex]\[ g(x) = x + 6 \][/tex]
Multiplying the two functions:
[tex]\[ (f \cdot g)(x) = (3x^2 + 25x + 42) \cdot (x + 6) \][/tex]
Apply the distributive property:
[tex]\[ (f \cdot g)(x) = 3x^2(x + 6) + 25x(x + 6) + 42(x + 6) \][/tex]
[tex]\[ (f \cdot g)(x) = 3x^3 + 18x^2 + 25x^2 + 150x + 42x + 252 \][/tex]
Combine like terms:
[tex]\[ (f \cdot g)(x) = 3x^3 + (18x^2 + 25x^2) + (150x + 42x) + 252 \][/tex]
[tex]\[ (f \cdot g)(x) = 3x^3 + 43x^2 + 192x + 252 \][/tex]
So, [tex]\( (f \cdot g)(x) = 3x^3 + 43x^2 + 192x + 252 \)[/tex].
### 4. Finding [tex]\(\frac{f}{g}\)[/tex] (quotient of [tex]\( f \)[/tex] and [tex]\( g \)[/tex]):
[tex]\[ \left( \frac{f}{g} \right)(x) = \frac{f(x)}{g(x)} \][/tex]
[tex]\[ f(x) = 3x^2 + 25x + 42 \][/tex]
[tex]\[ g(x) = x + 6 \][/tex]
So,
[tex]\[ \left( \frac{f}{g} \right)(x) = \frac{3x^2 + 25x + 42}{x + 6} \][/tex]
### Determining the Domains:
1. Domain of [tex]\((f+g)(x) = 3x^2 + 26x + 48\)[/tex]:
- This is a polynomial function, and polynomial functions are defined for all real numbers.
- Domain: [tex]\(\mathbb{R}\)[/tex] (all real numbers).
2. Domain of [tex]\((f-g)(x) = 3x^2 + 24x + 36\)[/tex]:
- This is also a polynomial function, defined for all real numbers.
- Domain: [tex]\(\mathbb{R}\)[/tex] (all real numbers).
3. Domain of [tex]\( (f \cdot g)(x) = 3x^3 + 43x^2 + 192x + 252 \)[/tex]:
- This is a polynomial function, defined for all real numbers.
- Domain: [tex]\(\mathbb{R}\)[/tex] (all real numbers).
4. Domain of [tex]\(\left( \frac{f}{g} \right)(x) = \frac{3x^2 + 25x + 42}{x + 6} \)[/tex]:
- The rational function is undefined when the denominator is zero.
- Set the denominator to zero to find the points to exclude:
[tex]\[ x + 6 = 0 \][/tex]
[tex]\[ x = -6 \][/tex]
Hence, the function is undefined at [tex]\( x = -6 \)[/tex].
- Domain: [tex]\(\mathbb{R} \setminus \{-6\}\)[/tex] (all real numbers except [tex]\( -6 \)[/tex]).
### Summary:
- [tex]\( (f+g)(x) = 3x^2 + 26x + 48 \)[/tex]
- Domain: [tex]\(\mathbb{R}\)[/tex]
- [tex]\( (f-g)(x) = 3x^2 + 24x + 36 \)[/tex]
- Domain: [tex]\(\mathbb{R}\)[/tex]
- [tex]\( (f \cdot g)(x) = 3x^3 + 43x^2 + 192x + 252 \)[/tex]
- Domain: [tex]\(\mathbb{R}\)[/tex]
- [tex]\( \left( \frac{f}{g} \right)(x) = \frac{3x^2 + 25x + 42}{x + 6} \)[/tex]
- Domain: [tex]\(\mathbb{R} \setminus \{-6\}\)[/tex]
Thank you for using this platform to share and learn. Don't hesitate to keep asking and answering. We value every contribution you make. IDNLearn.com provides the answers you need. Thank you for visiting, and see you next time for more valuable insights.
