Which of the following graphs is neither even nor odd?
A)
$f ( x ) = \frac { 4 x ^ { 2 } } { x ^ { 4 } + 1 }$
B) $f ( x ) = 8 x ^ { 3 } + 10 x ^ { 2 } + 1$
C) $f ( x ) = x ^ { 3 } - 9 x$

Question

Quizplus · Accepted Answer

To be even, a function must satisfy the condition $f(-x)=f(x)$, and to be odd, a function must satisfy the condition $f(-x)=-f(x)$. 
To check if a function is even, we can substitute $-x$ for $x$ and see if it is equivalent to $f(x)$. Similarly, to check if a function is odd, we can substitute $-x$ for $x$ and see if it is the negative of $f(x)$.
Let's apply these tests to each function:
A) $f(-x) = \frac{4(-x)^2}{(-x)^4+1} = \frac{4x^2}{x^4+1} = f(x)$, so this function is even.
B) $f(-x) = 8(-x)^3 + 10(-x)^2 + 1 = -8x^3 + 10x^2 + 1$, which is not equivalent to $f(x) = 8x^3 + 10x^2 + 1$, so this function is not even. 
$f(-x) = -8x^3 -10x^2 -1$ is not the negative of $f(x)$, so this function is also not odd. 
C) $f(-x) = (-x)^3 - 9(-x) = -x^3 + 9x$, which is not equivalent to $f(x) = x^3 - 9x$, so this function is not even. 
$f(-x) = -x^3 + 9x$ is the negative of $f(x)$, so this function is odd.
Thus, the only function that is neither even nor odd is B, which means its graph has no symmetry about the origin or the y-axis.

Which of the Following Graphs Is Neither Even nor Odd f(x)=4x2x4+1f ( x ) = \frac { 4 x ^ { 2 } } { x ^ { 4 } + 1 }f(x)=x4+14x2​

Which of the Following Graphs Is Neither Even nor Odd $f ( x ) = \frac { 4 x ^ { 2 } } { x ^ { 4 } + 1 }$