Question

How do you find all the numbers that must be excluded from the domain of the given rational expression $\dfrac{8}{{{x}^{2}}-4x}$?

Answer 1

Hint: We try to express the function and find the part which can make the expression invalid or undefined. We find the points on which the denominator part becomes equal to 0. Those points will be excluded from the domain of the expression $\dfrac{8}{{{x}^{2}}-4x}$.

Complete step by step answer:
We need to find the domain of the expression $\dfrac{8}{{{x}^{2}}-4x}$.
The condition being that the expression has to give a rational solution.
The numerator of the fraction $\dfrac{8}{{{x}^{2}}-4x}$ which is rational.
We only need to care about the denominator.
The denominator of the fraction $\dfrac{8}{{{x}^{2}}-4x}$ is a quadratic equation of $x$.
We know that the denominator of a fraction can never be 0.
So, the points which will be excluded from the domain of the expression $\dfrac{8}{{{x}^{2}}-4x}$ are the points which makes the denominator of the expression $\dfrac{8}{{{x}^{2}}-4x}$ equal to 0.
So, we need to find the value of $x$ for which ${{x}^{2}}-4x=0$.
We take $x$ common out of the equation ${{x}^{2}}-4x=0$ and form as a multiplication form.
Therefore, ${{x}^{2}}-4x=x\left( x-4 \right)=0$.
We get multiplication of two terms as 0 which gives at least one of them being equal to 0.
Either $x=0$ or $\left( x-4 \right)=0$. The solutions are $x=0,4$.
Therefore, these two points will make the expression $\dfrac{8}{{{x}^{2}}-4x}$ invalid.
The domain of the expression will be \[\mathbb{Q}\backslash \left\{ 0,4 \right\}\].

Note:
We need to remember that the denominator is solely responsible for the expression to be undefined. The value of 8 in the numerator changes nothing. Also, for any irrational value of x we will get the expression $\dfrac{8}{{{x}^{2}}-4x}$as irrational.