To determine the nature of the functions [tex]\((f \cdot g)(x)\)[/tex] and [tex]\((g \cdot g)(x)\)[/tex] given that [tex]\(f(x)\)[/tex] is even and [tex]\(g(x)\)[/tex] is odd, let's analyze their properties step-by-step.
### Definitions
1. Even Function:
- A function [tex]\( f(x) \)[/tex] is even if [tex]\( f(-x) = f(x) \)[/tex] for all [tex]\( x \)[/tex].
2. Odd Function:
- A function [tex]\( g(x) \)[/tex] is odd if [tex]\( g(-x) = -g(x) \)[/tex] for all [tex]\( x \)[/tex].
### Analysis
#### 1. [tex]\((f \cdot g)(x)\)[/tex]
To determine whether [tex]\((f \cdot g)(x)\)[/tex] is even, odd, or neither:
- By definition:
[tex]\[
(f \cdot g)(x) = f(x) \cdot g(x)
\][/tex]
- Evaluate [tex]\((f \cdot g)(-x)\)[/tex]:
[tex]\[
(f \cdot g)(-x) = f(-x) \cdot g(-x)
\][/tex]
- Since [tex]\( f(x) \)[/tex] is even:
[tex]\[
f(-x) = f(x)
\][/tex]
- Since [tex]\( g(x) \)[/tex] is odd:
[tex]\[
g(-x) = -g(x)
\][/tex]
- Substitute these properties:
[tex]\[
(f \cdot g)(-x) = f(-x) \cdot g(-x) = f(x) \cdot (-g(x)) = -f(x) \cdot g(x)
\][/tex]
- Notice that:
[tex]\[
(f \cdot g)(-x) = - (f \cdot g)(x)
\][/tex]
This tells us that [tex]\((f \cdot g)(x)\)[/tex] satisfies the definition of an odd function, since [tex]\((f \cdot g)(-x) = - (f \cdot g)(x)\)[/tex].
So,
[tex]\[
(f \cdot g)(x) \text{ is an } \textbf{odd} \text{ function}.
\][/tex]
#### 2. [tex]\((g \cdot g)(x)\)[/tex]
To determine whether [tex]\((g \cdot g)(x)\)[/tex] is even, odd, or neither:
- By definition:
[tex]\[
(g \cdot g)(x) = g(x) \cdot g(x)
\][/tex]
- Evaluate [tex]\((g \cdot g)(-x)\)[/tex]:
[tex]\[
(g \cdot g)(-x) = g(-x) \cdot g(-x)
\][/tex]
- Since [tex]\( g(x) \)[/tex] is odd:
[tex]\[
g(-x) = -g(x)
\][/tex]
- Substitute this property:
[tex]\[
(g \cdot g)(-x) = (-g(x)) \cdot (-g(x)) = g(x) \cdot g(x)
\][/tex]
- Notice that:
[tex]\[
(g \cdot g)(-x) = (g \cdot g)(x)
\][/tex]
This tells us that [tex]\((g \cdot g)(x)\)[/tex] satisfies the definition of an even function, since [tex]\((g \cdot g)(-x) = (g \cdot g)(x)\)[/tex].
So,
[tex]\[
(g \cdot g)(x) \text{ is an } \textbf{even} \text{ function}.
\][/tex]
### Summary:
- [tex]\( (f \cdot g)(x) \)[/tex] is an [tex]\(\textbf{odd}\)[/tex] function.
- [tex]\( (g \cdot g)(x) \)[/tex] is an [tex]\(\textbf{even}\)[/tex] function.
