Determine if each function is even, odd or neither. Show all work and include an explanation of any symmetry that exists.
A) y = x^4 + 4x^2
B) y = 3x^3 - x - 3
C) y = x^5 - x^3 + x
D)y = 3
To determine if a function is even, odd, or neither, we need to consider its symmetry.
1. A function is even if it is symmetric with respect to the y-axis, which means that replacing x with -x in the function gives the same result. To determine if a function is even, we need to check if f(x) = f(-x) for all x.
2. A function is odd if it is symmetric with respect to the origin (0,0), which means that replacing x with -x and y with -y in the function gives the same result. To determine if a function is odd, we need to check if f(x) = -f(-x) for all x.
Let's analyze each function:
A) y = x^4 + 4x^2
- Replacing x with -x: f(-x) = (-x)^4 + 4(-x)^2 = x^4 + 4x^2
- Comparing f(x) with f(-x): f(x) = f(-x)
- The function is even because f(x) = f(-x) for all x.
B) y = 3x^3 - x - 3
- Replacing x with -x: f(-x) = 3(-x)^3 - (-x) - 3 = -3x^3 + x - 3
- Comparing f(x) with -f(-x): f(x) = -f(-x)
- The function is odd because f(x) = -f(-x) for all x.
C) y = x^5 - x^3 + x
- Replacing x with -x: f(-x) = (-x)^5 - (-x)^3 + (-x) = -x^5 + x^3 - x
- Comparing f(x) with -f(-x): f(x) ≠ -f(-x)
- The function is neither even nor odd because f(x) is not equal to f(-x) or -f(-x) for all x.
D) y = 3
- Replacing x with -x: f(-x) = 3
- Comparing f(x) with f(-x): f(x) = f(-x)
- The function is even because f(x) = f(-x) for all x.
In summary:
A) Function A is even.
B) Function B is odd.
C) Function C is neither even nor odd.
D) Function D is even.