To determine whether each function [tex]\( (f \cdot g)(x) \)[/tex] and [tex]\( (g \cdot g)(x) \)[/tex] is even, odd, or neither given the properties of [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex], we need to use the definitions of even and odd functions.
### 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].
### Step-by-step solution:
#### 1. Determine if [tex]\( (f \cdot g)(x) \)[/tex] is even, odd, or neither:
Let’s consider the product [tex]\( (f \cdot g)(x) \)[/tex]:
[tex]\[
(f \cdot g)(x) = f(x) \cdot g(x)
\][/tex]
To determine the symmetry, compute [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 into the expression for [tex]\( (f \cdot g)(-x) \)[/tex]:
[tex]\[
(f \cdot g)(-x) = f(-x) \cdot g(-x) = f(x) \cdot (-g(x)) = - (f(x) \cdot g(x))
\][/tex]
Notice that:
[tex]\[
(f \cdot g)(-x) = - (f(x) \cdot g(x)) = - (f \cdot g)(x)
\][/tex]
This satisfies the definition of an odd function. Therefore,
[tex]\[
(f \cdot g)(x) \text{ is odd.}
\][/tex]
#### 2. Determine if [tex]\( (g \cdot g)(x) \)[/tex] is even, odd, or neither:
Now consider the product [tex]\( (g \cdot g)(x) \)[/tex]:
[tex]\[
(g \cdot g)(x) = g(x) \cdot g(x) = [g(x)]^2
\][/tex]
To determine the symmetry, compute [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 into the expression:
[tex]\[
(g \cdot g)(-x) = (-g(x)) \cdot (-g(x)) = [g(x)]^2
\][/tex]
Notice that:
[tex]\[
(g \cdot g)(-x) = [g(x)]^2 = (g \cdot g)(x)
\][/tex]
This satisfies the definition of an even function. Therefore,
[tex]\[
(g \cdot g)(x) \text{ is even.}
\][/tex]
### Summary:
1. [tex]\((f \cdot g)(x)\)[/tex] is odd.
2. [tex]\((g \cdot g)(x)\)[/tex] is even.
