Determine if the following function is even, odd, or neither: f(x) = -9x4 + 5x + 3
An even function is always symmetric about the y-axis, while an odd function is symmetric about the origin.
Answer: f(x) is clearly neither equal to f(-x) nor equal to -f(-x), so it is neither odd nor even function.
Go through the step by step process to understand the derivability of the nature of the function.
Given expression: f(x) = -9x4 + 5x + 3
Condition of even and odd functions:
1) For odd functions -
f(x) = -f(-x)
2) For even functions.
f(x) = f(-x)
3) If none of the above conditions hold true, then the function is neither odd nor even.
Let's check for the expression, f(x) = -9x4 + 5x + 3
Replace x by -x
f(-x) = 9(-x)4 + 5(-x) + 3
⇒ f(-x) = 9x4 - 5x + 3
Also we can see the value of -f(-x),
⇒ -f(-x) = -9x4 + 5x - 3
We can conclude that f(x) ≠ f(x) ≠ -f(-x)