Sure, let's solve this step by step.
Step 1: Define the given functions
We have [tex]\( f(x) = 3x^2 + 4x \)[/tex] and [tex]\( g(x) = -2x + 3 \)[/tex].
Step 2: Find [tex]\( f \cdot g \)[/tex]
To find [tex]\( (f \cdot g)(x) \)[/tex], we multiply the functions [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex]:
[tex]\[ (f \cdot g)(x) = f(x) \cdot g(x) \][/tex]
Substitute the given functions:
[tex]\[ (3x^2 + 4x) \cdot (-2x + 3) \][/tex]
Let's expand this product using the distributive property:
[tex]\[ (3x^2 + 4x) \cdot (-2x + 3) = 3x^2 \cdot -2x + 3x^2 \cdot 3 + 4x \cdot -2x + 4x \cdot 3 \][/tex]
Simplify each term:
[tex]\[ = -6x^3 + 9x^2 - 8x^2 + 12x \][/tex]
Combine like terms:
[tex]\[ = -6x^3 + (9x^2 - 8x^2) + 12x \][/tex]
[tex]\[ = -6x^3 + x^2 + 12x \][/tex]
Therefore:
[tex]\[ (f \cdot g)(x) = (3 - 2x)(3x^2 + 4x) \][/tex]
Step 3: Determine the domain of [tex]\( f \cdot g \)[/tex]
Both functions [tex]\(f(x)\)[/tex] and [tex]\(g(x)\)[/tex] are polynomials, and polynomials are defined for all real numbers. Therefore, the domain of [tex]\( f \cdot g \)[/tex] is all real numbers:
[tex]\[ \text{Domain of } f \cdot g: (-\infty, \infty) \][/tex]
Step 4: Find [tex]\( f - g \)[/tex]
Next, we find [tex]\( (f - g)(x) \)[/tex]. This is done by subtracting the function [tex]\( g(x) \)[/tex] from [tex]\( f(x) \)[/tex]:
[tex]\[ (f - g)(x) = f(x) - g(x) \][/tex]
Substitute the given functions:
[tex]\[ (f - g)(x) = (3x^2 + 4x) - (-2x + 3) \][/tex]
Distribute the negative sign:
[tex]\[ (f - g)(x) = 3x^2 + 4x + 2x - 3 \][/tex]
Combine like terms:
[tex]\[ (f - g)(x) = 3x^2 + (4x + 2x) - 3 \][/tex]
[tex]\[ (f - g)(x) = 3x^2 + 6x - 3 \][/tex]
Step 5: Determine the domain of [tex]\( f - g \)[/tex]
Again, both [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex] are polynomials, and polynomials are defined for all real numbers. Therefore, the domain of [tex]\( f - g \)[/tex] is all real numbers:
[tex]\[ \text{Domain of } f - g: (-\infty, \infty) \][/tex]
Summary:
1. [tex]\( (f \cdot g)(x) = (3 - 2x)(3x^2 + 4x) \)[/tex]
[tex]\[ \text{Domain of } f \cdot g: (-\infty, \infty) \][/tex]
2. [tex]\( (f - g)(x) = 3x^2 + 6x - 3 \)[/tex]
[tex]\[ \text{Domain of } f - g: (-\infty, \infty) \][/tex]
