Answer :
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]
