Select all of the equations below which are cubic functions.

[tex]\[
\begin{array}{cr}
y = x^2 + x + 1 & y = \frac{1}{5} x^3 \\
y = 4x^3 + x^2 + 2x + 5 & y = \frac{3}{x} \\
y = x - 4x^3 - 5 & y = \sqrt[3]{x} + 4
\end{array}
\][/tex]



Answer :

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].