Question

Find the domain of the function $g\left(x\right)=\sqrt[4]{x^2-6x}$.

Short answer

The domain of the function is $x\in \left(-\infty ,0\right]\cup \left[6,\infty \right)$.

Step-by-step solution

Step 1. By definition of domain of a function.

The given function is $g\left(x\right)=\sqrt[4]{x^2-6x}$. Now, by definition of domain of function, the domain of the function is defined for all real values of$\left(x\right)$ for which the function is defined.

Step 2. Evaluate domain of radical function.

From the function $g\left(x\right)=\sqrt[4]{x^2-6x}$, the domain is defined for all real numbers $\left(x\right)$for which the function $g\left(x\right)=\sqrt[4]{x^2-6x}$is defined. Since the function $g\left(x\right)=\sqrt[4]{x^2-6x}$is the radical function, therefore the domain of function is defined for all real numbers for which the expression $x^2-6x\ge 0$.

role="math" localid="1644590557338" $$\begin{gathered} \text{So,}{x}^2-6x\ge 0\text{\hspace{0.33em}gives,} \\ x\left(x-6\right)\ge 0 \\ x\le 0\text{or}x\ge 6 \end{gathered}$$

Hence the domain of the function is $x\in \left(-\infty ,0\right]\cup \left[6,\infty \right)$

Step 3. Description of steps.

The domain of the function is $x\in \left(-\infty ,0\right]\cup \left[6,\infty \right)$.