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Chapter 2: 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 is all real numbers.

Step by step solution

01

Denominator Equated to Zero

Set the denominator equal to zero.\[x^{2} + 64 = 0\]
02

Isolate the Variable x

Isolate the variable \(x\) by subtracting 64 from both sides. This gives us \[x^{2} = -64\]
03

Solve for x

Using the square root property to solve for \(x\), we get \(x = \pm \sqrt{-64} = \pm \sqrt{64 \cdot -1} = \pm 8i\). The solutions \(x = \pm 8i\) are imaginary.
04

Identify the Domain

Because there are no real roots, the original denominator \(x^{2} + 64\) will never equal zero and all real numbers are in the domain of the function.

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Most popular questions from this chapter

The equation for \(f\) is given by the simplified expression that results after performing the indicated operation. Write the equation for \(f\) and then graph the function. $$\frac{5 x^{2}}{x^{2}-4} \cdot \frac{x^{2}+4 x+4}{10 x^{3}}$$

If you are given the equation of a rational function, explain how to find the horizontal asymptote, if there is one, of the function's graph.

Will help you prepare for the material covered in the next section. Simplify: \(\frac{x+1}{x+3}-2\)

What is a rational function?

Determine whether each statement makes sense or does not make sense, and explain your reasoning. I'm solving a polynomial inequality that has a value for which the polynomial function is undefined.
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