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...
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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)`
`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.