# which of the following functions are odd?I. y=ln(x^3)II. y=|x^3|III. y=e^x^3a. noneb. II onlyc. I and IId. II and IIIe. I, II, and III

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Expert Answers

lemjay | Certified Educator

Note that a function is odd when: f(-x) = -y.

So, to determine which of the functions are odd, replace the in each function with -x.

I. `y=ln (x^3)`

`y=ln((-x)^3)`

`y=ln(-x^3)`

Note that in logarithm, a negative argument is not allowed. Hence, y=ln(x^3) is not an odd function..

II. `y =|x^3|`

`y=|(-x)^3|`

`y=|-x^3|`

Note that in absolute value,for any value inside the bracket, its resulting value is always positive.

`y=|x^3|`

Since it does not simplify to -y, then y=|x^3| is not an odd function.

|||. `y=e^(x^3)`

`y=e^((-x)^3)`

`y=e^(-x^3)`

`y=1/(e^(x^3))`

Also, this does not simplify to -y. Hence, y=e^(x^3) is not an odd function.

**Therefore among the given functions, none of them are odd.**