Question

Determine whether the equation$x=y^4$ defines$\left(y\right)$ as a function of$\left(x\right)$.

Short answer

The given equation $x=y^4$doesn’t define $\left(y\right)$ as a function of $\left(x\right)$.

Step-by-step solution

Step 1. Write down the given equation.

The given equation is $x=y^4$.

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

Solving the given equation,

$$\begin{gathered} x=y^4\mathrm{....}\left(\text{Given}\right) \\ \text{or,}{y}^4=x \\ {y}^2=\pm x\mathrm{....}\left(\text{Takesquareroot}\right) \end{gathered}$$

Step 3. Description of steps.

By simplifying the given equation, we obtain $y^2=\pm x$ from which it can be interpreted that for unique value of$\left(x\right)$ there exists two different values of $\left(y^2\right)$, one is positive and another negative. Since$\left(y^2\right)$ can-not be negative, so on taking positive value of$\left(y^2\right)$ again we can obtain two different values of$\left(y\right)$ while taking square-root. Therefore the given equation cannot be written in function form.