To determine if the given function is even, odd, neither, or both, we can analyze the symmetry of the graph.

Even Function:

A function is even if its graph is symmetric with respect to the y-axis. This means that for every point $(x, y)$ on the graph, there is a corresponding point $(-x, y)$ on the graph as well. Mathematically, $f(x) = f(-x)$.

Odd Function:

A function is odd if its graph is symmetric with respect to the origin. This means that for every point $(x, y)$ on the graph, there is a corresponding point $(-x, -y)$ on the graph as well. Mathematically, $f(x) = -f(-x)$.

Analysis of the Graph:

• Symmetry about the y-axis: The graph does not exhibit symmetry about the y-axis.
• Symmetry about the origin: The graph does not exhibit symmetry about the origin.

Since the graph does not show symmetry about the y-axis (even function) or the origin (odd function), the function is neither even nor odd.

Would you like more details or have any further questions?

Related Questions:

1. How do you determine the evenness or oddness of a function algebraically?
2. What are some examples of even and odd functions?
3. Can a function be both even and odd? If so, under what conditions?
4. How does the degree of a polynomial affect its symmetry properties?
5. What are the implications of a function being neither even nor odd?
6. How would the graph change if the function were odd?
7. How can transformations (like translations) affect the symmetry of a graph?
8. What does the end behavior of this graph suggest about the function's degree?

Tip:

When analyzing the symmetry of a graph, always check the symmetry with respect to the y-axis for even functions and with respect to the origin for odd functions.