The following condition will help determine if a function is even or odd or neither even nor odd
[tex]If\text{ }f\mleft(x\mright)=f\mleft(-x\mright)\text{ ,then the function is even}[/tex][tex]If,\text{ }f\mleft(x\mright)=-f\mleft(-x\mright),\text{ then the function is odd}[/tex][tex]\text{If the function does not satisfy the two conditions, then the function is neither even nor odd}[/tex]Let us determine the nature of the function given
[tex]\begin{gathered} f(x)=-2x^3+5x^2-3x+1 \\ f(-x)=-2(-x)^3+5(-x)^2-3(-x)+1 \\ f(-x)=-2(-x^3)+5(x)^2-3(-x)+1 \\ f(-x)=2x^3+5x^2+3x+1 \end{gathered}[/tex][tex]\begin{gathered} \text{since} \\ f(x)\ne f(-x), \\ \text{then the function is not even} \end{gathered}[/tex][tex]\begin{gathered} f(x)=-2x^3+5x^2-3x+1 \\ f(-x)=2x^3+5x^2+3x+1 \\ -f(-x)=-(2x^3+5x^2+3x+1) \\ -f(-x)=-2x^3-5x^2-3x-1 \end{gathered}[/tex][tex]\begin{gathered} \sin ce, \\ f(x)\ne-f(-x) \\ \text{then, the function is not odd} \end{gathered}[/tex]Based on the above findings, it can be observed that the function is not even and it is not odd,
Hence, the function is neither even nor odd, OPTION C