Question

Find the domain of the function $f\left(x\right)=\frac{x^4}{x^2+x-6}$.

Short answer

The domain$\left(D_f\right)$ of the function is the set of all real numbers$\left(x\right)$ such that$\left(x\right)$ belongs to real except $-3,2$.

Step-by-step solution

Step 1. By definition of domain of a function.

The given function is $f\left(x\right)=\frac{x^4}{x^2+x-6}$. 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. By definition of domain of rational function.

From the rational function $f\left(x\right)=\frac{x^4}{x^2+x-6}$, the domain is defined for all real numbers $\left(x\right)$for which the denominator is non-zero.

role="math" localid="1644579100031" $$\begin{gathered} x^2+x-6\ne 0\text{\hspace{0.17em}gives} \\ \left(x+3\right)\left(x-2\right)\ne 0 \\ x\ne -3,2 \end{gathered}$$

Hence the domain of the function is$x\in R-\{-3,2\}$

Step 3. Description of steps.

The domain of the function is $x\in R-\{-3,2\}$.