Let's go through each part of the question step-by-step.
1. Expression for [tex]\((f \cdot g)(x)\)[/tex]:
[tex]\[ (f \cdot g)(x) = f(x) \cdot g(x) \][/tex]
Given the functions:
[tex]\[ f(x) = 5x^3 \][/tex]
[tex]\[ g(x) = x^2 \][/tex]
Substitute for [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex]:
[tex]\[ (f \cdot g)(x) = (5x^3) \cdot (x^2) \][/tex]
[tex]\[ (f \cdot g)(x) = 5x^5 \][/tex]
2. Expression for [tex]\((f+g)(x)\)[/tex]:
[tex]\[ (f+g)(x) = f(x) + g(x) \][/tex]
Substitute for [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex]:
[tex]\[ (f+g)(x) = 5x^3 + x^2 \][/tex]
3. Evaluate [tex]\((f-g)(-2)\)[/tex]:
[tex]\[ (f-g)(x) = f(x) - g(x) \][/tex]
First, determine [tex]\( f(-2) \)[/tex] and [tex]\( g(-2) \)[/tex]:
[tex]\[ f(-2) = 5(-2)^3 = 5(-8) = -40 \][/tex]
[tex]\[ g(-2) = (-2)^2 = 4 \][/tex]
Then, evaluate [tex]\((f-g)(-2)\)[/tex]:
[tex]\[ (f-g)(-2) = f(-2) - g(-2) = -40 - 4 = -44 \][/tex]
Hence, the results are:
[tex]\[ (f \cdot g)(x) = 5x^5 \][/tex]
[tex]\[ (f+g)(x) = 5x^3 + x^2 \][/tex]
[tex]\[ (f-g)(-2) = -44 \][/tex]
