How do you find the domain of f(x)= (8x)/((x-1)(x-2)) ?

Mar 30, 2018

$x \in \mathbb{R} , x \ne 1 , 2$

Explanation:

$f \left(x\right) \text{ is defined for all values of x except values which}$
$\text{make f(x) undefined}$

$\text{the denominator of f(x) cannot be zero as this would make}$
$\text{f(x) undefined. Equating the denominator to zero and }$
$\text{solving gives the values that x cannot be}$

$\text{solve } \left(x - 1\right) \left(x - 2\right) = 0$

$\Rightarrow x = 1 , x = 2 \leftarrow \textcolor{red}{\text{are excluded values}}$

$\text{domain is } x \in \mathbb{R} , x \ne 1 , 2$

$\left(- \infty , 1\right) \cup \left(2 , + \infty\right) \leftarrow \textcolor{b l u e}{\text{in interval notation}}$
graph{(8x)/((x-1)(x-2)) [-10, 10, -5, 5]}

Mar 30, 2018

Since the given function is a rational function, look where is the denominator is equal to zero. ($\frac{8 x}{0}$is not defined)
(x-1)(x-2)=0 if ${x}_{1} = 1$ or ${x}_{2} = 2$
The domain of $f \left(x\right)$ is : $\mathbb{R} - \left\{1 , 2\right\}$ real number except 1 and 2