Answer :
To solve the problem where we need to find [tex]\( f(x) \cdot g(x) \)[/tex] and its domain, let's follow these step-by-step:
#### Step 1: Define the functions
Given the functions:
[tex]\[ f(x) = 5x + 3 \][/tex]
[tex]\[ g(x) = x^2 - 6x + 5 \][/tex]
#### Step 2: Perform the function operation [tex]\( f(x) \cdot g(x) \)[/tex]
To find [tex]\( f(x) \cdot g(x) \)[/tex], we need to multiply the two functions together:
[tex]\[ f(x) \cdot g(x) = (5x + 3)(x^2 - 6x + 5) \][/tex]
#### Step 3: Expand and simplify the expression
Now let's expand the expression:
[tex]\[ (5x + 3)(x^2 - 6x + 5) = 5x \cdot x^2 + 5x \cdot (-6x) + 5x \cdot 5 + 3 \cdot x^2 + 3 \cdot (-6x) + 3 \cdot 5 \][/tex]
[tex]\[ = 5x^3 - 30x^2 + 25x + 3x^2 - 18x + 15 \][/tex]
Combine like terms:
[tex]\[ = 5x^3 + (-30x^2 + 3x^2) + (25x - 18x) + 15 \][/tex]
[tex]\[ = 5x^3 - 27x^2 + 7x + 15 \][/tex]
So, the simplified form is:
[tex]\[ f(x) \cdot g(x) = 5x^3 - 27x^2 + 7x + 15 \][/tex]
#### Step 4: Determine the domain of [tex]\( f(x) \cdot g(x) \)[/tex]
To find the domain of [tex]\( f(x) \cdot g(x) \)[/tex], consider the domains of [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex]:
- [tex]\( f(x) = 5x + 3 \)[/tex] is a linear function, which is defined for all real numbers.
- [tex]\( g(x) = x^2 - 6x + 5 \)[/tex] is a quadratic function, which is also defined for all real numbers.
Since both functions are polynomials, and polynomials are defined for all real numbers, the domain of [tex]\( f(x) \cdot g(x) \)[/tex] will also be all real numbers.
#### Answer:
[tex]\[ f(x) \cdot g(x) = 5x^3 - 27x^2 + 7x + 15 \][/tex]
The domain of [tex]\( f(x) \cdot g(x) \)[/tex] is the set of all real numbers.
Thus, the correct answer is:
C. The domain of [tex]\( f(x) \cdot g(x) \)[/tex] is the set of all real numbers.
#### Step 1: Define the functions
Given the functions:
[tex]\[ f(x) = 5x + 3 \][/tex]
[tex]\[ g(x) = x^2 - 6x + 5 \][/tex]
#### Step 2: Perform the function operation [tex]\( f(x) \cdot g(x) \)[/tex]
To find [tex]\( f(x) \cdot g(x) \)[/tex], we need to multiply the two functions together:
[tex]\[ f(x) \cdot g(x) = (5x + 3)(x^2 - 6x + 5) \][/tex]
#### Step 3: Expand and simplify the expression
Now let's expand the expression:
[tex]\[ (5x + 3)(x^2 - 6x + 5) = 5x \cdot x^2 + 5x \cdot (-6x) + 5x \cdot 5 + 3 \cdot x^2 + 3 \cdot (-6x) + 3 \cdot 5 \][/tex]
[tex]\[ = 5x^3 - 30x^2 + 25x + 3x^2 - 18x + 15 \][/tex]
Combine like terms:
[tex]\[ = 5x^3 + (-30x^2 + 3x^2) + (25x - 18x) + 15 \][/tex]
[tex]\[ = 5x^3 - 27x^2 + 7x + 15 \][/tex]
So, the simplified form is:
[tex]\[ f(x) \cdot g(x) = 5x^3 - 27x^2 + 7x + 15 \][/tex]
#### Step 4: Determine the domain of [tex]\( f(x) \cdot g(x) \)[/tex]
To find the domain of [tex]\( f(x) \cdot g(x) \)[/tex], consider the domains of [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex]:
- [tex]\( f(x) = 5x + 3 \)[/tex] is a linear function, which is defined for all real numbers.
- [tex]\( g(x) = x^2 - 6x + 5 \)[/tex] is a quadratic function, which is also defined for all real numbers.
Since both functions are polynomials, and polynomials are defined for all real numbers, the domain of [tex]\( f(x) \cdot g(x) \)[/tex] will also be all real numbers.
#### Answer:
[tex]\[ f(x) \cdot g(x) = 5x^3 - 27x^2 + 7x + 15 \][/tex]
The domain of [tex]\( f(x) \cdot g(x) \)[/tex] is the set of all real numbers.
Thus, the correct answer is:
C. The domain of [tex]\( f(x) \cdot g(x) \)[/tex] is the set of all real numbers.
