For these functions, determine which ones are NEITHER even nor odd.
A) f(x)=3x^2+|x|
B) f(x)= 3x^3+x^2+x-4
C) f(x)= 1/2x
D) f(x)= 2x^5-x^3+2x
E) f(x)= 4x+1/x
F) F(x)= 2x^4+x^2-x+2
Enter the letter of each function separated with a comma.
How can you tell which ones are even or odd?
To determine if a function is even, odd, or neither, we need to understand the properties of even and odd functions:
1. Even function: A function f(x) is even if, for every x in its domain, f(-x) = f(x). In other words, if the function is symmetric about the y-axis.
2. Odd function: A function f(x) is odd if, for every x in its domain, f(-x) = -f(x). In other words, if the function is symmetric about the origin.
Now let's examine each function and determine if it is even, odd, or neither.
A) f(x) = 3x^2 + |x|
To determine if it is even or odd, we substitute -x for x in the function:
f(-x) = 3(-x)^2 + |-x| = 3x^2 + |x|
Since f(-x) = f(x), it is an even function.
B) f(x) = 3x^3 + x^2 + x - 4
Again, substitute -x for x in the function:
f(-x) = 3(-x)^3 + (-x)^2 + (-x) - 4 = -3x^3 + x^2 - x - 4
Since f(-x) is not equal to f(x) and f(-x) is not equal to -f(x), it is neither even nor odd.
C) f(x) = 1/2x
Substituting -x for x:
f(-x) = 1/2(-x) = -1/2x
Since f(-x) is not equal to f(x) and f(-x) is also not equal to -f(x), it is neither even nor odd.
D) f(x) = 2x^5 - x^3 + 2x
Substituting -x for x:
f(-x) = 2(-x)^5 - (-x)^3 + 2(-x) = -2x^5 - x^3 - 2x
Since f(-x) is not equal to f(x) and f(-x) is not equal to -f(x), it is neither even nor odd.
E) f(x) = 4x + 1/x
Substituting -x for x:
f(-x) = 4(-x) + 1/(-x) = -4x - 1/x
Since f(-x) is not equal to f(x) and f(-x) is not equal to -f(x), it is neither even nor odd.
F) f(x) = 2x^4 + x^2 - x + 2
Substituting -x for x:
f(-x) = 2(-x)^4 + (-x)^2 - (-x) + 2 = 2x^4 + x^2 + x + 2
Since f(-x) is not equal to f(x) and f(-x) is not equal to -f(x), it is neither even nor odd.
Based on the analysis above, the functions B, C, D, E, and F are neither even nor odd.