Question

In Exercises \(21-30\) , determine whether the function is even, odd, or neither. Try to answer without writing anything (except the answer). $$y=x+x^{3}$$

Short answer

The function \(y=x+x^{3}\) is odd.

Step-by-step solution

Identify the Function

The given function is \(y=x+x^{3}\).

Testing for Even Function

Replace \(x\) with \(-x\) in the function \(y=x+x^{3}\). The new function becomes \(y=-x-(-x)^{3}=-x-x^{3}\). This does not match the original function, so it's not even.

Testing for Odd Function

Replace \(x\) with \(-x\) in the function \(y=-x-x^{3}\) that we derived from the previous step which is the negated version of the original function. The new function becomes \(-y=-(-x)-(-x)^{3}=x+x^{3}\). This matches exactly with our original function, so it's odd.