Question

6x - 2y³ = -4
x\ty
-2\t6
-1\t7
0\t8
1\t9
2\t10
2\t11
none of the above

Explanation:

Step1: Define function criteria

A function is a relation where each input ($x$-value) has exactly one output ($y$-value). We check each option against this rule.

Step2: Analyze the equation $6x - 2y^3 = -4$

Rearrange to solve for $y$:

$$\begin{align*} -2y^3&=-4-6x\\ y^3&=3x+2\\ y&=\sqrt[3]{3x+2} \end{align*}$$

Each $x$ gives exactly one real $y$, so this is a function.

Step3: Analyze the mapping diagram

Input $a$ maps to 0, 1, 2, 3 (multiple outputs). This violates the function rule, so it is not a function.

Step4: Analyze the table

Input $x=2$ maps to $y=10$ and $y=11$ (multiple outputs). This violates the function rule, so it is not a function.

Step5: Analyze the vertical graph

A vertical line $x=5$ has infinitely many $y$-values for one $x$-value. This violates the function rule, so it is not a function.

Answer:

$\boldsymbol{6x - 2y^3 = -4}$ (this is the only valid function among the options)