Question

How do you find the domain of $f\left( x \right) = \dfrac{{4{x^2} - 9}}{{{x^2} + 5x + 6}}$?

Answer 1

Hint: This is a simple question based on function and its domain set and codomain set. As we know that the domain of $f\left( x \right)$ is the set of all those real numbers for which $f\left( x \right)$ is meaningful. For this, determine the domain of the real function $f\left( x \right)$ by finding all those real numbers for which the expression for $f\left( x \right)$ or the formula for $f\left( x \right)$ assumes real values only.

Complete step by step answer:
Given function: $f\left( x \right) = \dfrac{{4{x^2} - 9}}{{{x^2} + 5x + 6}}$
We have to find the domain of a given function.
First understand the concept of domain, then determine the domain of given function.
Since we know that the domain of the real function $f\left( x \right)$ is the set of all those real numbers for which the expression for $f\left( x \right)$ or the formula for $f\left( x \right)$ assumes real values only. In other words, the domain of $f\left( x \right)$ is the set of all those real numbers for which $f\left( x \right)$ is meaningful.
Here, $f\left( x \right) = \dfrac{{4{x^2} - 9}}{{{x^2} + 5x + 6}}$
Clearly, $f\left( x \right)$ is a rational function of $x$ as $\dfrac{{4{x^2} - 9}}{{{x^2} + 5x + 6}}$ is a rational expression in $x$. We observe that $f\left( x \right)$ assumes real values for all $x$ except for all those of $x$ for which ${x^2} + 5x + 6 = 0$.
We are using the split middle term method.
${x^2} + 5x + 6 = 0$ writing the middle term in terms of $2x,3x$.
${x^2} + 2x + 3x + 6 = 0$
Now, taking the common
$x\left( {x + 2} \right) + 3\left( {x + 2} \right) = 0$
$ \Rightarrow \left( {x + 2} \right)\left( {x + 3} \right) = 0$
$ \Rightarrow x = - 2, - 3$
Hence, Domain ($f$) $ = \mathbb{R} - \left\{ { - 2, - 3} \right\}$.

Therefore, the domain of given function is $\mathbb{R} - \left\{ { - 2, - 3} \right\}$.

Note: In above question, we can determine the domain of a given question by simply drawing the graph of the function.

Therefore, the domain of a given function is $\mathbb{R} - \left\{ { - 2, - 3} \right\}$.