Question

Find the domain of the function

\(f\left( x \right) = \frac{{x + 4}}{{{x^2} - 9}}\)

Short answer

The domain of the function \(f\left( x \right) = \frac{{x + 4}}{{{x^2} - 9}}\)is\(\left( { - \infty ,\; - 3} \right) \cup \left( { - 3,\;3} \right) \cup \left( {3,\;\infty } \right)\).

Step-by-step solution

Describe the domain

The domain is the set of all values at which the given function is defined.

Find the domain of the function\(f\left( x \right) = \frac{{x + 4}}{{{x^2} - 9}}\)

Find the values of \(x\) at which the function is not defined.

\(\begin{array}{c}{x^2} - 9 = 0\\{x^2} - {\left( 3 \right)^2} = 0\\\left( {x - 3} \right)\left( {x + 3} \right) = 0\\x = 3,\; - 3\end{array}\)

Therefore, the domain of the given function will be all the values of \(x\) except \(3\), and\( - 3\).

Therefore, the domain of the function \(f\left( x \right) = \frac{{x + 4}}{{{x^2} - 9}}\)is\(\left( { - \infty ,\; - 3} \right) \cup \left( { - 3,\;3} \right) \cup \left( {3,\;\infty } \right)\).