Question

How do you find all the values that make the expression undefined: $\dfrac{3{{z}^{2}}+z}{18z+6}$?

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 $\dfrac{3{{z}^{2}}+z}{18z+6}$.

Complete step by step answer:
We need to find the domain of the expression $\dfrac{3{{z}^{2}}+z}{18z+6}$.
The condition being that the expression has to give a defined solution.
The numerator of the fraction $\dfrac{3{{z}^{2}}+z}{18z+6}$ which is rational. It is a quadratic equation of $z$.
We only need to care about the denominator.
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 $\dfrac{3{{z}^{2}}+z}{18z+6}$ are the points which makes the denominator of the expression $\dfrac{3{{z}^{2}}+z}{18z+6}$ equal to 0.
So, we need to find the value of $z$ for which $18z+6=0$.
We subtract 6 from both sides.
$\begin{align}
  & 18z+6-6=0-6 \\
 & \Rightarrow 18z=-6 \\
\end{align}$.
We now divide both sides with 18 and get
$\begin{align}
  & \dfrac{18z}{18}=\dfrac{-6}{18} \\
 & \Rightarrow z=-\dfrac{1}{3} \\
\end{align}$
Therefore, this point $z=-\dfrac{1}{3}$ will make the expression $\dfrac{3{{z}^{2}}+z}{18z+6}$ invalid.
The domain of the expression will be \[\mathbb{R}\backslash \left\{ -\dfrac{1}{3} \right\}\].

Note:
We need to remember that the denominator is solely responsible for the expression to be undefined. The value of any $z$ in the numerator of $\dfrac{3{{z}^{2}}+z}{18z+6}$ changes nothing. Also, for any irrational value of $z$ we will get the expression $\dfrac{3{{z}^{2}}+z}{18z+6}$ as irrational.