Question

The function f(x) = -2x³ +5x²-3x+1 isA.evenB.oddC.neither even nor oddD. symmetric about the y-axis

Answer 1

The following condition will help determine if a function is even or odd or neither even nor odd

`If\text{ }f\mleft(x\mright)=f\mleft(-x\mright)\text{ ,then the function is even}`
`If,\text{ }f\mleft(x\mright)=-f\mleft(-x\mright),\text{ then the function is odd}`
`\text{If the function does not satisfy the two conditions, then the function is neither even nor odd}`

Let us determine the nature of the function given

`\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}`
`\begin{gathered} \text{since} \\ f(x)\\e f(-x), \\ \text{then the function is not even} \end{gathered}`
`\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}`
`\begin{gathered} \sin ce, \\ f(x)\\e-f(-x) \\ \text{then, the function is not odd} \end{gathered}`

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