To determine which of the following functions are cubic functions, we need to understand that a cubic function in terms of [tex]\( x \)[/tex] has the highest power of [tex]\( x \)[/tex] to be 3.
Let’s analyze each function step by step:
### Option A:
[tex]\[ 8y^3 = 2x^2 + 2x - 27 \][/tex]
First, solve for [tex]\( y \)[/tex]:
[tex]\[ y^3 = \frac{2x^2 + 2x - 27}{8} \][/tex]
This function primarily shows [tex]\( y \)[/tex] in terms of [tex]\( x \)[/tex]. However, upon realizing the highest power of [tex]\( x \)[/tex] on the right-hand side of the equation is 2, this is not a cubic function in [tex]\( x \)[/tex], it’s a cubic function in [tex]\( y \)[/tex].
### Option B:
[tex]\[ y = 2x^2 - 5x + 3 \][/tex]
Looking at the powers of [tex]\( x \)[/tex]:
The highest power of [tex]\( x \)[/tex] here is 2.
This indicates the function is quadratic in terms of [tex]\( x \)[/tex], not cubic.
### Option C:
[tex]\[ y = 10 - 2x \][/tex]
Similarly, observe the powers of [tex]\( x \)[/tex]:
The highest power of [tex]\( x \)[/tex] here is 1.
This indicates the function is linear in terms of [tex]\( x \)[/tex], not cubic.
### Option D:
[tex]\[ y = x^2(x - 2) \][/tex]
Let’s expand and simplify this equation:
[tex]\[ y = x^2 \cdot x - x^2 \cdot 2 \][/tex]
[tex]\[ y = x^3 - 2x^2 \][/tex]
Here, the highest power of [tex]\( x \)[/tex] is 3.
This indicates the function is cubic in terms of [tex]\( x \)[/tex].
### Conclusion:
Out of the given functions, the only function that is cubic with respect to [tex]\( x \)[/tex] is:
[tex]\[ \boxed{y = x^2(x - 2)} \][/tex] (Option D)
