Final answer:
To determine if functions are even, odd, or neither, use the conditions y(x) = y(-x) for even functions and y(x) = -y(-x) for odd functions. The function g(x) = |x-3| is neither even nor odd. The function g(x) = x·x² simplifies to x³, which is an odd function.
Step-by-step explanation:
To analyze symmetry algebraically and determine whether each function is even, odd, or neither, we can apply specific criteria for even and odd functions. An even function satisfies the condition y(x) = y(-x), which implies that the function is symmetric about the y-axis. On the other hand, an odd function satisfies the condition y(x) = -y(-x), indicating that the function is symmetric about the origin.
For the function g(x) = |x-3|, to determine if it's even or odd, we can substitute -x for x:
- g(-x) = |-x-3| = |-(x+3)| = |x+3|
As we can see, g(x) is neither even nor odd because g(-x) does not equal g(x) nor does it equal -g(x).
The function g(x) = x·x² simplifies to g(x) = x³, which is an odd function. We verify this by applying the odd function test:
Since g(x) = -g(-x), this function is odd.
Even functions multiplied by other even functions remain even, and the same applies for odd functions. However, an odd function multiplied by an even function results in an odd function. This is important in various mathematical and physical contexts, such as expectation-value calculations and the study of wave functions in quantum mechanics.