Problem 6 - Algebra and Trigonometry

Question

Find the domain of each rational function. $$ h(x)=\frac{x+8}{x^{2}-64} $$

Step-by-Step Solution

Answer

The domain of the function \( h(x)=\frac{x+8}{x^{2}-64} \) is all real numbers except \( x = 8 \) and \( x = -8 \).

Step 1: Identify the Denominator

For the given rational function \( h(x)=\frac{x+8}{x^{2}-64} \), the denominator is \( x^{2}-64 \). We need to find the values of \( x \) that make this denominator equal to zero.

Step 2: Solve the Equation for Zero

We set \( x^{2}-64 = 0 \) and solve for \( x \). This will give \( x^{2}=64 \), and solving further gives us two possible values, \( x=8 \) and \( x=-8 \)

Step 3: Exclude Values that make Denominator Zero from Domain

The values \( x=8 \) and \( x=-8 \) make the denominator zero and hence, they are not included in the domain. So, the domain of the rational function \( h(x)=\frac{x+8}{x^{2}-64} \), is all real numbers except \( x = 8 \) and \( x = -8 \).