Step 1: Evaluate the function for each element in the domain

Given $X = \{-1, 0, 1\}$ and $f(x) = 2x^4$. We calculate the image of each element:

For $x = -1$: $f(-1) = 2(-1)^4 = 2(1) = 2$

For $x = 0$: $f(0) = 2(0)^4 = 2(0) = 0$

For $x = 1$: $f(1) = 2(1)^4 = 2(1) = 2$

Step 2: Check for One-One (Injective) property

A function is one-one if distinct elements have distinct images. Here, $f(-1) = 2$ and $f(1) = 2$. Since two distinct elements in the domain map to the same element in the codomain, the function is not one-one.

Step 3: Check for Onto (Surjective) property

A function is onto if the range equals the codomain. The range of $f$ is $\{0, 2\}$. The codomain $Y$ is given as $\{0, 2\}$. Since Range = Codomain, every element in $Y$ has a pre-image in $X$. Therefore, the function is onto.