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
even if f(-x) = f(x)
odd if f(-x) = -f(x)
A even
B neither
C odd
D even
all even functions have axis of symmetry at y=0
all odd functions are radially symmetric about (0,0)
a product of an even and an odd function is odd.
y=3x^3 - x - 3 = 3x(x^2 - 1/3) - 3
so, since x^2-1/3 is even, and 3x is odd, 3x(x^2 - 1/3) is odd.
y is radially symmetric about (0,-3)
To determine whether a function is even, odd, or neither, we need to examine its symmetry properties.
1) A function is even if it satisfies the equation f(x) = f(-x) for all values of x.
2) A function is odd if it satisfies the equation f(x) = -f(-x) for all values of x.
3) A function is neither even nor odd if it does not satisfy either of the above equations for all values of x.
Now let's analyze each function:
A) y = x^4 + 4x^2
To determine if this function is even, odd, or neither, we'll substitute -x for x in the equation and see if it remains the same:
f(-x) = (-x)^4 + 4(-x)^2
= x^4 + 4x^2
Since f(x) = f(-x), the function is even. It exhibits symmetry about the y-axis.
B) y = 3x^3 - x - 3
Let's substitute -x for x and check if the equation holds:
f(-x) = 3(-x)^3 - (-x) - 3
= -3x^3 + x - 3
Since -f(-x) = -3x^3 + x + 3, which is not equal to f(x), this function is neither even nor odd. It does not exhibit any particular symmetry.
C) y = x^5 - x^3 + x
Let's apply the same process:
f(-x) = (-x)^5 - (-x)^3 + (-x)
= -x^5 + x^3 - x
Since -f(-x) = -x^5 + x^3 - x, which is equal to f(x), this function is odd. It exhibits symmetry about the origin (0,0).
D) y = 3
Regardless of the value of x, the function remains constant at y = 3. It is symmetric about the y-axis and is considered an even function.
In summary:
A) The function y = x^4 + 4x^2 is even. It exhibits symmetry about the y-axis.
B) The function y = 3x^3 - x - 3 is neither even nor odd. It does not exhibit any special symmetry.
C) The function y = x^5 - x^3 + x is odd. It exhibits symmetry about the origin (0,0).
D) The function y = 3 is even. It exhibits symmetry about the y-axis.