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Chapter 3: Problem 8

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

Short Answer

Expert verified
The domain of the function \(f(x)=\frac{x+8}{x^{2}+64}\) is the set of all real numbers.

Step by step solution

01

Understand the Function

The function given is \(f(x)=\frac{x+8}{x^{2}+64}\). The domain of this function is all x-values for which the function is defined, meaning where the denominator does not equal zero.
02

Set the denominator equal to zero

To find when the function is undefined, set the denominator equal to zero and solve for x. Therefore, solve the equation \(x^{2}+64=0\). Subtract 64 from both sides to obtain \(x^{2}=-64\).
03

Solve for x

In real numbers, the square of a number cannot be negative. Therefore, the equation \(x^{2}=-64\) has no real solutions.
04

Find the Domain

Since there is no real number that make the denominator equal to zero, the function is defined for all real numbers. Therefore, the domain of this function is the set of all real numbers.

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