Let's analyze the nature of the functions step by step.
Given:
- [tex]\( f(x) \)[/tex] is an even function.
- [tex]\( g(x) \)[/tex] is an odd function.
We need to determine whether the following functions are even, odd, or neither:
1. [tex]\((f \cdot g)(x)\)[/tex]
2. [tex]\((g \cdot g)(x)\)[/tex]
### Step 1: Analyze [tex]\((f \cdot g)(x)\)[/tex]
To determine the nature of the function [tex]\((f \cdot g)(x)\)[/tex], we will consider:
[tex]\[ (f \cdot g)(x) = f(x) \cdot g(x) \][/tex]
For a function [tex]\( h(x) \)[/tex] to be even, it must satisfy:
[tex]\[ h(x) = h(-x) \][/tex]
For a function [tex]\( h(x) \)[/tex] to be odd, it must satisfy:
[tex]\[ h(x) = -h(-x) \][/tex]
#### Check if [tex]\((f \cdot g)(x)\)[/tex] is even:
[tex]\[ (f \cdot g)(-x) = f(-x) \cdot g(-x) \][/tex]
Since [tex]\( f(x) \)[/tex] is even, we know that:
[tex]\[ f(-x) = f(x) \][/tex]
Since [tex]\( g(x) \)[/tex] is odd, we know that:
[tex]\[ g(-x) = -g(x) \][/tex]
Putting these together:
[tex]\[ (f \cdot g)(-x) = f(-x) \cdot g(-x) = f(x) \cdot (-g(x)) = -f(x) \cdot g(x) = -(f \cdot g)(x) \][/tex]
We see that:
[tex]\[ (f \cdot g)(-x) = -(f \cdot g)(x) \][/tex]
This satisfies the definition of an odd function. Therefore,
[tex]\[ (f \cdot g)(x) \text{ is odd} \][/tex]
### Step 2: Analyze [tex]\((g \cdot g)(x)\)[/tex]
To determine the nature of the function [tex]\((g \cdot g)(x)\)[/tex], we will consider:
[tex]\[ (g \cdot g)(x) = g(x) \cdot g(x) = (g(x))^2 \][/tex]
We need to check if this is an even or odd function.
#### Check if [tex]\((g \cdot g)(x)\)[/tex] is even:
[tex]\[ (g \cdot g)(-x) = g(-x) \cdot g(-x) = (g(-x))^2 \][/tex]
Since [tex]\( g(x) \)[/tex] is odd:
[tex]\[ g(-x) = -g(x) \][/tex]
Therefore:
[tex]\[ (g \cdot g)(-x) = (-g(x))^2 = g(x)^2 \][/tex]
We see 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]
### Conclusion:
- [tex]\((f \cdot g)(x) = \)[/tex] neither
- [tex]\((g \cdot g)(x) = \)[/tex] even
