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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lemjay | High School Teacher | (Level 3) Senior Educator

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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)`



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

II. `y =|x^3|`



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


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

|||. `y=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.