To determine whether the function f(x) = 3x² + 1 is even, odd, or neither, we need to evaluate the function using its definition.
1. **Definition of Even Function:** A function is called even if for every x in the domain of the function, f(-x) = f(x). This means that the graph of the function is symmetric with respect to the y-axis.
2. **Definition of Odd Function:** A function is called odd if for every x in the domain of the function, f(-x) = -f(x). The graph of an odd function is symmetric with respect to the origin.
Now, let’s calculate f(-x):
f(-x) = 3(-x)² + 1
= 3(x²) + 1 (since (-x)² = x²)
= 3x² + 1
Now we can compare f(-x) and f(x):
f(x) = 3x² + 1
Since we found that f(-x) = f(x), we conclude that:
The function is even.
In conclusion, the function f(x) = 3x² + 1 is even because it satisfies the condition f(-x) = f(x). Therefore, it exhibits symmetry about the y-axis.