Question

Find the domain of the function $f\left(x\right)=\frac{1}{3x-6}$.

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 2.

Step-by-step solution

Step 1. By definition of domain of a function.

The given function is $f\left(x\right)=\frac{1}{3x-6}$. 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 $f\left(x\right)=\frac{1}{3x-6}$, the domain is defined for all real numbers $\left(x\right)$for which the denominator is non-zero.

$$\begin{gathered} 3x-6\ne 0\text{\hspace{0.17em}gives} \\ x\ne 2 \end{gathered}$$

Hence the domain of the function is$x\in R-\left\{2\right\}$

Step 3. Description of steps.

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