Chapter 1, Problem 50RE

To determine

To find:Thedomain of the function $h\left(x\right)=\frac{x}{x^2-x-6}$ .

Answer to Problem 50RE

The domain of the function $h\left(x\right)=\frac{x}{x^2-x-6}$ is all real number except at $-2,3$ .

Explanation of Solution

Given information:

The given function is $h\left(x\right)=\frac{x}{x^2-x-6}$ .

Calculation:

The domain of the function is $x$ values for which function have a definite real value. In given function we have real solutions if the denominator have non zero values. So, domain of function is defined by region given as.

  $\begin{array}{c}{x}^2-x-6\ne 0\\ x^2-3x+2x-6\ne 0\\ \left(x+2\right)\left(x-3\right)\ne 0\end{array}$

  $x\ne -2$ and $x\ne 3$ are the roots of the above quadratic equation. So, domain of the function is all real number except at $-2,3$ .

Graph for the given function is shown in figure (1).

  

Figure (1)

From the above graph in figure (1), shows that the function does not have a definite value at $-2,3$ .

Therefore, the domain of the function $h\left(x\right)=\frac{x}{x^2-x-6}$ is all real number except at $-2,3$ .