Q. 7.20 - Intermediate Algebra

Question

Find the domain of $R\left(x\right)=\frac{4x^2-16x}{8x^2-16x-64}$.

Step-by-Step Solution

Answer

The domain of $R\left(x\right)$ is all real numbers, where $x\ne 4$ and $x\ne -2.$

Step 1 . Given information

The given expression is$R\left(x\right)=\frac{4x^2-16x}{8x^2-16x-64}$

Step 2 . Concept used

The domain of the rational function is all real numbers except the values that make the denominator equal to zero.

Procedure to find the domain:

• Set the denominator equal to zero.
• Solve the equation.
• The domain is all real numbers except the values found in the previous step.

Step 3 . Find the domain

$R\left(x\right)=\frac{4x^2-16x}{8x^2-16x-64}$

Set the denominator to zero and solve

$$\begin{gathered} 8x^2-16x-64=0 \\ {x}^2-2x-8=0 \text{ Write out factor 8} \\ {x}^2-4x+2x-8=0 \text{ Expand the middle term} \\ x(x-4)+2(x-4)=0 \text{Write out common elements} \\ (x-4)(x+2)=0 \text{Write out common term} \\ x-4=0 \text{ or }x+2=0 \text{ Solve for} x \\ x=4 \text{or }x=-2 \end{gathered}$$

Hence, the domain is all real numbers except at $x=4,-2.$