To determine which of the given equations are cubic functions, we need to identify which equations represent polynomials with a degree of 3. A cubic function is characterized by the highest exponent of the variable [tex]\( x \)[/tex] being 3. Let's examine each equation in detail:
1. [tex]\( y = x^2 + x + 1 \)[/tex]
- The highest power of [tex]\( x \)[/tex] is 2.
- This is not a cubic function.
2. [tex]\( y = \frac{1}{5} x^3 \)[/tex]
- The highest power of [tex]\( x \)[/tex] is 3.
- This is a cubic function.
3. [tex]\( y = 4 x^3 + x^2 + 2 x + 5 \)[/tex]
- The highest power of [tex]\( x \)[/tex] is 3.
- This is a cubic function.
4. [tex]\( y = \frac{3}{x} \)[/tex]
- This can be rewritten as [tex]\( y = 3x^{-1} \)[/tex].
- The highest power of [tex]\( x \)[/tex] is -1, not 3.
- This is not a cubic function.
5. [tex]\( y = x - 4 x^3 - 5 \)[/tex]
- The highest power of [tex]\( x \)[/tex] is 3.
- This is a cubic function.
6. [tex]\( y = \sqrt[3]{x} + 4 \)[/tex]
- This can be rewritten as [tex]\( y = x^{1/3} + 4 \)[/tex].
- The highest power of [tex]\( x \)[/tex] is [tex]\( 1/3 \)[/tex], not 3.
- This is not a cubic function.
Thus, the equations that are cubic functions are:
[tex]\[
\frac{1}{5} x^3, \quad 4 x^3 + x^2 + 2 x + 5, \quad x - 4 x^3 - 5
\][/tex]
These correspond to the following indices in the original list:
1. [tex]\( y = \frac{1}{5} x^3 \)[/tex] (2nd equation)
2. [tex]\( y = 4 x^3 + x^2 + 2 x + 5 \)[/tex] (3rd equation)
3. [tex]\( y = x - 4 x^3 - 5 \)[/tex] (5th equation)
Therefore, the indices of the cubic functions are [tex]\([1, 2, 4]\)[/tex].
