Chapter 2, Problem 9RE

To determine

To find the domain of the function $f\left(x\right)=\frac{x}{x^2-9}$ .

Answer to Problem 9RE

All real numbers $x$ except $x=-3,3$ .

Explanation of Solution

Given:

A function is given as

  $f\left(x\right)=\frac{x}{x^2-9}$

Concept used:

If domain is not specified for a function, the set of largest real number should be considered where the function is defined all over.

For rational function denominator should be non-zero for its domain.

Calculation:

Domain of $f\left(x\right)=\frac{x}{x^2-9}$ is the values ofx for which denominator is non-zero.

That is,

  $\begin{array}{l}{x}^2-9\ne 0\\ \left(x-3\right)\left(x+3\right)\ne 0\\ x-3\ne 0\&x+3\ne 0\\ x\ne 3\&x\ne -3\end{array}$

Hence the domain of $f\left(x\right)=\frac{x}{x^2-9}$ is the set of all real numbers $x$ except $x=-3,3$ .