Respuesta :
we have the function
[tex]f\mleft(x\mright)=x^3-2x[/tex]part 1
Verify if the function is symmetric with respect to the y-axis
A graph is symmetric with the y-axis
if
f(x)=f(-x)
so
[tex]\begin{gathered} f(-x)=(-x)^3-2(-x) \\ f(-x)=-x^3+2x \\ therefore \\ f(x)\ne\text{ f\lparen-x\rparen} \end{gathered}[/tex]The graph is not symmetric with respect to the y-axis
part 2
Verify if the function is symmetric with respect to the x-axis
A graph is symmetric with the x-axis
If
f(x)=-f(x)
so
[tex]\begin{gathered} -f(x)=-(x^3-2x) \\ -f(x)=-x^3+2x \\ therefore \\ f(x)\ne-f(x) \end{gathered}[/tex]The graph is not symmetric with respect to the x-axis
Part 3
Verify if the function is symmetric with respect to the origin
A graph is symmetric with the origin
if
f(x)=-f(-x)
so
[tex]\begin{gathered} -f(-x)=-[(-x)^3-2(-x)] \\ -f(-x)=-[-x^3+2x] \\ -f(-x)=x^3-2x \\ therefore \\ f(x)=-f(-x) \end{gathered}[/tex]therefore
The answer is
