# Question #4e641

Sep 29, 2016

This is an odd function.

#### Explanation:

To check if a function $f \left(x\right)$ is odd or even (or neither) you have to calculate $f \left(- x\right)$ and compare the calculated value with $f \left(x\right)$.

• If $f \left(- x\right) = f \left(x\right)$ then $f \left(x\right)$ is even,

• if $f \left(- x\right) = - f \left(x\right)$ then $f \left(x\right)$ is odd.

Here we have:

$f \left(x\right) = {x}^{3} + 7 x$

$f \left(- x\right) = {\left(- x\right)}^{3} + 7 \cdot \left(- x\right)$

$f \left(- x\right) = - {x}^{3} - 7 x$

We see that the value is an opposite expression to $f \left(x\right)$.

$f \left(- x\right) = - f \left(x\right)$, so the function is odd

Other way to see if the function is even or odd is to look at the function's graph.

• if the graph is symetrical according to $Y$ axis then the function is even.Example: $y = {x}^{2} + 2$

graph{x^2+2 [-10, 10, -5, 5]}

• If the graph is symetrical according to the origin $\left(0 , 0\right)$ then it is odd. Example: ${x}^{3} + 7 x$

graph{x^3+7x [-3, 3, -50, 50]}