Question

Determine whether the equation$x=y^3$defines$\left(y\right)$ as a function of$\left(x\right)$.

Short answer

The given equation $x=y^3$defines $\left(y\right)$as a function of $\left(x\right)$.

Step-by-step solution

Step 1. Write down the given equation.

The given equation is $x=y^3$.

Step 2. Solve the given equation for (y) in terms of (x).

Solving the given equation,

$$\begin{gathered} x=y^3\mathrm{....}\left(\text{Given}\right) \\ \text{or,}{y}^3=x \\ y=\sqrt[3]{x}\mathrm{....}\left(\text{Takecube-root}\right) \\ y=x^{\frac{1}{3}} \end{gathered}$$

Step 3. DESCRIPTION OF STEPS.

By simplifying the given equation, we obtain$y=\sqrt[3]{x}$ from which it can be interpreted that for unique value of$\left(x\right)$ there exists a unique $\left(y\right)$. There for the given equation represents$\left(y\right)$ as a function of $\left(x\right)$, which can also be written in function form as $f\left(x\right)={\left(x\right)}^{\frac{1}{3}}$.