It appears that the given piecewise function definition is not in a standard form, and there seems to be some confusion in the notations and symbols. Typically, a piecewise function is defined using conditions that specify different expressions for different intervals of the input variable. Let's try to rewrite the function in a standard form before solving it.
Here's a potential correct interpretation:
Given
[tex]\[ a(n) = \begin{cases}
4 & \text{if} \; \pi < 0 \\
\frac{1}{2} & \text{if} \; n = 0 \\
n & \text{if} \; n = 4
\end{cases} \][/tex]
Now, let's consider standard piecewise notation and analyze each condition in turn.
### Condition 1
[tex]\[ 4 \;\; \text{if} \;\; \pi < 0 \][/tex]
Since [tex]\(\pi \approx 3.14159\)[/tex], it's always greater than 0. Thus, this condition is never true. So, this part does not contribute to the definition of [tex]\( a(n) \)[/tex].
### Condition 2:
[tex]\[ \frac{1}{2} \;\; \text{if} \;\; n = 0 \][/tex]
This means [tex]\( a(n) = \frac{1}{2} \)[/tex] when [tex]\( n = 0 \)[/tex].
### Condition 3:
[tex]\[ n \;\; \text{if} \;\; n = 4 \][/tex]
This means [tex]\( a(n) = 4 \)[/tex] when [tex]\( n = 4 \)[/tex].
Since [tex]\(\pi < 0\)[/tex] is never true, we can drop that part and rewrite the function more cleanly as:
[tex]\[ a(n) = \begin{cases}
\frac{1}{2} & \text{if} \; n = 0 \\
4 & \text{if} \; n = 4
\end{cases} \][/tex]
To illustrate this function graphically, we focus on the specific points defined by the piecewise conditions:
1. At [tex]\( n = 0 \)[/tex], [tex]\( a(0) = \frac{1}{2} \)[/tex].
2. At [tex]\( n = 4 \)[/tex], [tex]\( a(4) = 4 \)[/tex].
For all other values of [tex]\(n\)[/tex], [tex]\(a(n)\)[/tex] is not defined according to the given piecewise definition.
Therefore, the graph will consist of two points:
- A point at (0, 0.5).
- A point at (4, 4).
There will not be any lines connecting these points since the function is not defined between these values.
To summarize, the graph of this piecewise function will show:
1. A single point at the coordinates (0, 0.5).
2. A single point at the coordinates (4, 4).
The rest of the graph will remain empty because these are the only values for which [tex]\( a(n) \)[/tex] is defined.
